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Presentador:
Dr. Julio Quintana-UPR Mayagüez
RECURSOS PARA FACILITADORES DEL PROGRAMA DE MATEMÁTICAS DEL
DEPARTAMENTO DE EDUCACIÓN DE PUERTO RICO (DEPR)
Materiales CRAIM
DIDÁCTICA DE LA MATEMÁTICA
Materiales CRAIM
DIDÁCTICA MATEMÁTICA
Trigonometría
Presentador:
Dr. Julio Quintana
UPR, Mayagüez
RECURSOS PARA FACILITADORES DEL PROGRAMA DE MATEMÁTICAS
DEL DEPARTAMENTO DE EDUCACIÓN DE PUERTO RICO (DEPR)
Dr. Jorge M. López Director CRAIM, UPR Río Piedras
© Todos los derechos reservados
Publicaciones CRAIM
2012
Trigonometría
Didáctica matemática
Escuela secundaria
Recursos para facilitadores del PM del DEPR
Director Dr. Jorge M. López
Universidad de Puerto Rico, recinto de Río Piedras
Introducción
Los materiales de la serie Didáctica Matemática para los facilitadores del Programa de
Matemática (PM) y los maestros del nivel secundario del Departamento de Educación de Puerto
Rico (DEPR) se han confeccionado para atender temas medulares del nuevo currículo a la luz de
los documentos de Estándares curriculares y las Expectativas por grado. Esta serie recopila temas
identificados, por los maestros y facilitadores de matemática, como prioritarios en la matemática
del nivel secundario y se presentan con la profundidad que requiere el nuevo currículo. Estos
materiales de la serie Didáctica Matemática, cubren los siguientes temas específicos:
1) Funciones, modelación matemática, graficación. Transformación de funciones y sus
gráficas. Funciones polinómicas, racionales, logarítmicas y exponenciales.
2) Los modelos funcionales sinusoidales, leyes de senos y cosenos. Ondas sinusoidales y
el principio de superposición.
3) Empleo de las funciones como modelos matemáticos, graficación y geometría
analítica.
4) Probabilidad y estadística. Distribuciones binomial y normal. Regresión lineal y la
recta de cuadrados mínimos. Coeficiente de correlación.
El material didáctico presentado atiende especialmente temas prioritarios contenidos en los
siguientes cursos:
I. Matemática en acción (grado 10, primer curso)
II. Aventuras matemáticas (grado 10, segundo curso)
III. Funciones y modelos (grado 11, primer curso)
IV. Matemáticas contemporáneas (grado 11, segundo curso)
V. Precálculo (grado 12)
Esperamos que estos recursos sean de utilidad y hagan más efectiva la labor de los facilitadores
del Programa de matemáticas y los maestros del nivel secundario del DEPR.
Jorge M. López
Catedrático de Matemática
Universidad de Puerto Rico
Río Piedras
9 de septiembre de 2012
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(a)
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..........................................................................................
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..........................................................................................................................................................................................................................................................
..
....................................................................................................................................................................................................................... ..
.......................................................................................................................................................................................................................
(b)
2! "#$$#% $# '()*)# )($ &(/'(.+, *.)* #), ,. x! 4#& 1/0%#& ., (&+-. # (& #$#! !"!" #$%&'() *+,-&'&./0+, $)#$ %&'( &% *+$ *$* *(,-& %. %&'/0*- 0% .( '1*2-$-,%'13( (&- *($0- ( (0( 4$2/.- (2/0- α *%1'-& $5,%1-& 6/% ..(,(,-& .(& !"#$%& ' ()#$#*+' ( !& 0%. 4$2/.-7 8(1( .( 0%9$* *+$*$* *(.: &/;-$%,-& 6/% $/%&'1- 4$2/.- α %& /$- 0% .-& 4$2/.-& (2/0-& 0% /$ '1*4$2/.-1% '4$2/.-: %& 0% *1: /$ '1*4$2/.- 6/% '*%$% /$ 4$2/.- 1% '-7 <(& 1(=-$%& '1*2-$-,>'1* (& &-$&%*&7 ?(@ '1%& ;1*$ *;(.%&: @ .(& -'1(& '1%& &-$ .-& 1% 3;1- -& 0% .(& ;1*,%1(&7 A%9$*1%,-& %. #&%$#: %. &%$#: .( '!$)%$'%: @ &/& 1% 3;1- -&: .( &% !$'%: .( #&% !$'%: @ .( #'!$)%$'%0%. 4$2/.-: .(& 6/% &*,B-.*=(,-& ;-1 #&(α)- &%$(α): '!$(α)- &% (α)- & (α) @ #'(α)1%&;% '*C(,%$'%7.%/$( (0$ 12313 !" α !# $%&'#( "&')( *%)* ")( !% !# ,*&'*!%-! -.*$%&'#(.! -$%&'#(
......................................................................................................................................................................................................................................................................................................................................................................................... ............
a
b
c
αA B
C
cos(α) :=b
csen(α) :=
a
ctan(α) :=
a
b
sec(α) :=c
bcsc(α) :=
c
acot(α) :=
b
a
!"!" #$%&'() *#+,&'&-.*#+/$) !" !"#$%& ()*#"+#$ %&'() *) +(),-./ -.0102/ 0+-).&/. 3() 40, .05/+), -.&'/+/67-.& 0, *) 40 *)9+& &:+ *);<)+*)+ ,/406)+-) *)4 =+'(4/ > +/ *) 40, 6)*&*0, ),<) ?9 0, *)4 -.&=+'(4/ .) -=+'(4/3() (,06/,@ >0 3() (043(&). /-./ -.&=+'(4/ .) -=+'(4/ 3() -)+'0 (+ =+'(4/ 0'(*/&'(04 0 α ), ,)6)20+-) 04 -.&=+'(4/ *0*/@ </. 4/ -0+-/@ 40, .05/+), )+-.) 4/, 40*/, ,).=+&'(04), 0 40, .05/+), )+-.) 4/, 40*/, *)4 -.&=+'(4/ /.&'&+04$A$ B,(046)+-) .) /.*06/, 40, *)9+& &/+), *) 40 ,&'(&)+-) 60+).0C <0.0 (+ =+'(4/ 0'(*/α )+ (+ -.&=+'(4/ .) -=+'(4/D ! #$%&# '% α %$ !( )(*+& %&,)% %! (,%,# ('-( %&,% (! .&/0!# α - !( 123#,%&0$(4 ED ! $%&# '% α %$ !( )(*+& %&,)% %! (,%,# #30%$,# (! .&/0!# α - !( 123#,%&0$(4 ED 5( ,(&/%&,% '% α %$ !( )(*+& %&,)% %! (,%,# #30%$,# - %! (,%,# ('-( %&,% ( α4 E,-#./0) 12+2+1+ F+ )4 -.&=+'(4/ .) -=+'(4/ &+*& 0*/ 0 /+-&+(0 &:+@ G0440. )4 H04/. *)4 /,)+/@ )4 ,)+/ > 40 -0+')+-) *)4 =+'(4/ α$I0.0 .),</+*). 0 ),0, <.)'(+-0, +) ),&-06/, 40 6)*&*0 *)40 G&</-)+(,0@ 40 3() 04 (40.)6/, (,0+*/ )4 J)/.)60 *)I&-='/.0,$ J)+)6/, 3() c2 = 52 + 32 = 34K ), *) &. 3()
c =√34$I/. 4/ -0+-/@
cos(α) =5√34
, sen(α) =3√34
, tan(α) =3
5.,-#./0) 12+2+2+ I0.0 )4 -.&=+'(4/ *)4 )2)6<4/ 0+-).&/.@ 04 (40. )4 /,)+/@ )4 ,)+/ > 40-0+')+-) *)4 =+'(4/ 0'(*/ β LH). 40 9'(.0M$
cos(β) =3√34
, sen(β) =5√34
, tan(β) =5
3.,-#./0) 12+2+3+ %01&)+*/ 3() )4 /,)+/ *) &).-/ =+'(4/ ), &'(04 0 1/3@ G0440. )4 H04/. *)4,)+/ *)4 =+'(4/$6%$#!0 2+&4 I0.0 </*). .),</+*). +) ),&-0.)6/, (+ -.&=+'(4/ .) -=+'(4/ 3() /+-)+'0 (+=+'(4/ 0'(*/ (>/ /,)+/ ,)0 1/3$ B+/ *) -04), -.&=+'(4/, ,) 6(),-.0 0 /+-&+(0 &:+@ */+*)G)6/, 44060*/ α 04 =+'(4/$
! !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+.........................................................................................................................................
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................................................................................................................................................................ ............
1
3 x
α"#$%&% '(% %) +$%$, +-.+ %/$% + α 01-% 2 01%/$3+& '(% )+ 415,$%/(&+ 01-% 67 8)+0+/-, x+ )+ 0%-1-+ -%) +$%$, ,5(%&$,9 $%/%0,& 5,3 :1$;<,3+& '(%x2 + 12 = 32,-% -,/-% x2 = 89 , &%+9 x =
√87 :,3 ), $+/$,9 sen(α) = √8
37=)(&$3+3%0,& ),& (&,& -% ,&%/, . &%/, ,/ (/,& %>%05),& 0(. &105)%&7 ?/ +-+ (/, -%),& %>%05),& 1/-1 +0,& @+),3%& 5+3+ %) ,&%/, . %) &%/, -% (/ ;/<(),9 . 5,3 +4,3+ /, /,&53%, (5+3%0,& + %3 + -% ,0, ,A$%/%3 %&,& @+),3%&9 .+ '(% ), 105,3$+/$% +'(B %& 0,&$3+3 ,0, &% (&+/ @+),3%& ,/, 1-,& -% ,&%/, . &%/, 5+3+ 3%&,)@%3 53,A)%0+&7 !"#$%& '()()*) C/ +A)% -% )+3<, D! 51%& @+ -%&-% %) &(%), 4+&$+ )+ 5(/$+ -% (/ 5,&$%9-% 0+/%3+ $+) '(% %) ;/<(), α E,30+-, 5,3 %) +A)% . %) &(%), $1%/% (/ ,&%/, 1<(+) + 0.6 .(/ &%/, 1<(+) + 0.87 FG '(H -1&$+/ 1+ -% )+ A+&% -%) 5,&$% %&$; )+ 5(/$+ -%) +A)% +$+-+ +)&(%),I FJ(;) %& )+ +)$(3+ -%) 5,&$%I
......................................................................................................................................................................................................................................................................................................................................................................................... ............
20
αA B
C
!"#$% '()* K13+/-, )+ L<(3+ /,$+0,& '(% $%/%0,& (/ $31;/<(), 3% $;/<(), ,/ 415,$%/(&+1<(+) +) +A)% . (., +$%$, ,5(%&$, +) ;/<(), α %& 1<(+) +) 5,&$%7C&+/-, )+ /,$+ 1#/ -% )+ L<(3+9 $%/%0,& '(% ,/, %0,& c . '(%3%0,& -%$%301/+3 b M)+-1&$+/ 1+ -% )+ 5(/$+ %/ %) &(%), -%) +A)% + )+ A+&% -%) 5,&$%N . a M)+ +)$(3+ -%) 5,&$%N7G4,3+ A1%/9 5,3 )+ -%L/1 1#/ -% ,&%/,9 $%/%0,& '(% b
c= cos(α)9 -% -,/-% ,A$%/%0,& '(%9
b = c · cos(α) = 20 · 0.6 = 12 51%&7 G/;),<+0%/$%9 ac= sen(α)9 -% -,/-% &% ,A$1%/% '(%9
a = c · sen(α) = 20 · 0.8 = 16 51%&. !"#$%& '()()+) C/ 5,&$% (.+ +)$(3+ &% -%& ,/, %9 5%3, (.+ &,0A3+ %/ 1%3$, 0,0%/$,-%) -B+ $1%/% (/+ ),/<1$(- -% 2D 0%$3,& . %) ;/<(), -% %)%@+ 1#/ -%&-% )+ 5(/$+ -% )+ &,0A3++) $,5% -%) 5,&$% %& (/ ;/<(), α $+) '(% cos(α) ≈ 0.8944 . sen(α) ≈ 0.44727 FO%3; 5,&1A)% +) ()+3 )+ +)$(3+ -%) 5,&$%I !"#$% '()* 8+ &1<(1%/$% L<(3+ 0(%&$3+ (/ %&'(%0+ -% )+ &1$(+ 1#/
!"!" #$%&'() *#+,&'&-.*#+/$) !".........................................................................................................................................
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.......
.............
.............
.............
.............
.............
.............
.........
12α
A B
C
...........................................................
............. !"#$#$ %&'()& *+,-& $./)0) 1+) *20 %+$/20 2&&)0%2$4')$/)0 - *- %+$/- 4) *- 02(5&-6 - *- 5-0)4)* %20/) 7 - *- %+$/- 4)* %20/)6 4)/)&('$-$ +$ /&'8$,+*2 &) /8$,+*2 42$4) α )0 )* 8$,+*22%+)0/2 -* %20/)9 #* %&25*)(- 2$0'0/) )$ :-**-& a 0-5')$42 1+) b = 12 7 2$2 ')$42 )* 20)$2 7 0)$2 4)* 8$,+*2 α9 ;2(2 0) 2$2 ) 1+) )* 0)$2 %2& 4)<$' '.$ )0 a/c6 0) /')$) 1+)a
c= sen(α) =⇒ a = c sen(α). =>?@2 1+) '(%*' - 1+) *- -*/+&- 4)* %20/) =a? 0)&8 2$2 '4- +-$42 2$2A -(20 *- ()4'4- 4)*- :'%2/)$+0- c9B-&- 2(%+/-& 4' :- ()4'4- +0-(20 *2 0',+')$/)9 C- 1+)
cos(α) =b
c=⇒ c =
b
cos(α).D+0/'/+7)$42 )* E-*2& 25/)$'42 %-&- c )$ *- &)*- '.$ =>?6 25/)$)(20 1+)
a = c sen(α) =a
cos(α)sen(α) = b
sen(α)
cos(α).D+0/'/+7)$42 )$ *- )F%&)0'.$ *20 E-*2&)0 2$2 '4206 /)$)(20 1+)
a = bsen(α)
cos(α)≈ 12
0.4472
0.8944= 6 ()/&20. !"#$%& '()()*) G-**-& )* 8&)- 4)* 0',+')$/) /&'8$,+*2 &) /8$,+*26 0-5')$42 1+) b = 156
c = 25 7 sen(α) ≈ 0.7669................................................................................................................................................................................................
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A B
C
D 25
15
α !"#$% '()* ;2(2 0-5)(20 1+) %24)(20 -* +*-& )* 8&)- 4) +$ /&'8$,+*2 ()4'-$/) *- H.&(+*-I&)- =1
2J-0) ·K*/+&- =
1
2· 15 · a7 *- 5-0) )0 2$2 '4- =',+-* - L"?6 $) )0'/-(20 -E)&',+-& *- -*/+&- a9 ;2(2 )* /&'8$,+*2 )0&) /8$,+*26 7 2$2 )(20 /-$/2 b 2(2 )* sen(α) 0) /')$) 1+)
a
25= sen(α),
!" !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+#$ #%&#$'a = 25 sen(α) = 25(0.766) = 19.15()$*%' $+ ,-$. /$#0#. $12-$. =
1
2ba =
1
2(15)(19.15) ≈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�#% /.-. #$9$-40&.- #0 A%1 5.+%-$1B J$ %-#$4%1 :)$ +. 9-0*%&%4$9-@. $1 )&. 0$& 0.50$<. 40$&9-.1 :)$ +.1 .+ )+.#%-.1 1%& )& .5.& $ -$ 0$&9$BK& $19. 1$ 0?& 5$-$4%1 %4% .+ )+.- +%1 5.+%-$1 #$ +.1 -.6%&$1 7-0*%&%489-0 .1 #$ .+*)&%1,&*)+%1 $1/$ 0.+$1 4$#0.&9$ %&10#$-. 0%&$1 *$%489-0 .1 104/+$1B !"#$%& '()()*) L$9$-40&.- $+ %1$&%' $+ 1$&% = +. 9.&*$&9$ #$ )& ,&*)+% #$ M◦B !"#$% '()* L0;)<$4%1 )& 9-0,&*)+% -$ 9,&*)+% %& )& ,&*)+% .*)#% #$ M ◦B
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .
.
.
.
.
.......45◦A B
C
N%4% 1.;$4%1 :)$ +%1 #%1 ,&*)+%1 .*)#%1 1%& %4/+$4$&9.-0%1' 1$ 90$&$ :)$ +. 4$#0#. #$+,&*)+% β $1 90 − 45 = 45◦B N%& +)04%1 $&9%& $1 :)$ $+ 9-0,&*)+% $1 01?1 $+$1B O%- +% 9.&9%'1) %1$&% = 1$&% 90$&$& $+ 4014% 5.+%-B N%4% +.1 -.6%&$1 &% #$/$&#$& #$ +.1 4$#0#.1 #$+%1 +.#%1' /%#$4%1 .10*&.- ).+:)0$- 5.+%- . +%1 .9$9%1 a = b' #0*.4%1 PB O%- O09,*%-.1'9$&$4%1 :)$ c2 = 12 + 12' #$ 4.&$-. :)$ c =√2B O%- +% 9.&9%'
cos(45◦) =b
c=
1√2=
√2
2.
!"!" #$%&'() *#+,&'&-.*#+/$) !"sen(45◦) =
a
c=
1√2=
√2
2.#$%&'() *$+ ()+,*-'.$+ '/-)(0$()+ $/ '2,)**$+ $3-)/0.$+ $/ ,/' '* ,*'.$('4 5$-) 2,) '.06)()/ 0' .) *' '* ,*'.$('7 '2,8 9)%$+ $3-)/0.$ ,/ ()+,*-'.$ !" $% &'(' *$+ :'*$()+ .) $+)/$ ; +)/$4 <+'/.$ *$ '/-)(0$(7 -)/)%$+ '.)%=+ 2,)
tan(45◦) =a
b=
1
1= 1.&' ()*% +,-,-.- >)-)(%0/'( )* $+)/$7 )* +)/$ ; *' -'/?)/-) .) ,/ =/?,*$ .) @!◦4 !"#$% '()* A,+2,)%$+ ,/ -(0=/?,*$ () -=/?,*$ .$/.) ,/$ .) +,+ =/?,*$+ +)' .) @!◦4 B*=/?,*$ .) @!◦)+ ,/ =/?,*$ )+&) 0'*7 ;' 2,) 3 · 60 = 1804 #$%$ )/ ,/ -(0=/?,*$ )2,0*=-)($ '.' =/?,*$ %0.) @!◦7 ,+'()%$+ ,/ -(0=/?,*$ )2,0*=-)($ &'(' $3-)/)( *$+ :'*$()+ .)+)'.$+4CD'+) *' +0?,0)/-) E?,(' .$/.) 9)%$+ .03,F'.$ ,/ -(0=/?,*$ )2,0*=-)($4 G0/ &D(.0.' .)?)/)('*0.'.7 &$.)%$+ +,&$/)( 2,) *$+ *'.$+ .)* -(0=/?,*$ %0.)/ H4 G0 .0:0.0%$+ )* -(0=/?,*$-('I'/.$ *' 30+) -(0I .) ,/$ .) *$+ =/?,*$+
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
A B
C
D
60◦60◦$3-)/)%$+ .$+ -(0=/?,*$+ () -=/?,*$+ 0.D/-0 $+4 G0 -('3'F'%$+ $/ ,/$ .) )**$+ J:)' *' E?,('KB/-$/ )+7 c = 17 b = 1/2 ; ,+'/.$ L0-=?$('+ 9'**'%$+ )* :'*$( .) a
a2 + b2 = c2
=⇒ a2 = c2 − b2
=⇒ a2 = 12 − (1
2)2 = 1− 1
4=
3
4
=⇒ a =
√3
2.M,)?$ -)/)%$+ 2,)
cos(60◦) =b
c=
1/2
1=
1
2.
sen(60◦) =a
c=
√3/2
1=
√3
2.
tan(60◦) =a
b=
√3/2
1/2=√3.
!" !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+ !"#$%& '()()*) #$%$&'()*& $+ -.$)-/ $+ .$)- 0 +* %*)1$)%$ 2$ 3) 4)13+- 2$ 5!◦6 !"#$% '()* 7%(+(8*)2- $+ '(.'- %&(4)13+-/ .$ 9$ 2$ +* :13&* ;3$cos(30◦) =
a
c=
√3
2
sen(30◦) =b
c=
1
2
tan(30◦) =b
a=
1/2√3/2
=1√3=
√3
3<$.3'('-. +-. &$.3+%*2-. $) +* .(13($)%$ %*=+*630◦ 45◦ 60◦ -.$)- √3
2
√2
2
1
2.$)- 1
2
√2
2
√3
2>3*2&- ?@6?A B*+-&$. 2$ -.$)- 0 .$)- 2$ 5!◦/ C◦0 D!◦ !"!"!" #$%&'($)$%* +,(-.&./0',( )* 23&$)/%&')4%*E* $FG$&($) (* 2$ *+ 3+*& 3)* &*8H) %&(1-)-'I%&( * $) %I&'()-. 2$ -%&* '3$.%&* ;3$ J*0 3)*.$&($ 2$ &$+* (-)$. *+1$=&*( *. $)%&$ +*. &*8-)$. %&(1-)-'I%&( *.6 K) $.%* .$ (H) 9$&$'-. *+L13)*. &$+* (-)$. 2$ (13*+2*2 $)%&$ +*. 2(M$&$)%$. &*8-)$. %&(1-)-'I%&( *. ;3$/ G-& .$& 94+(2*.G*&* 3*+;3($& 4)13+-/ .$ ++*'*) +,- ./"012/,/"- 3425&0៾ ,- 890/,#"01,%"-6E*. G&('$&*. %&$. (2$)%(2*2$. M3)2*'$)%*+$. .-) +*. .(13($)%$.:&1,05"01"; <" ,01" 0 :&-" ,01"6cot(α) =
1
tan(α), sec(α) =
1
cos(α), csc(α) =
1
sen(α)#$ +*. 2$:)( (-)$. ;3$ *G*&$ $) $) +* .$ (H) *)%$&(-& .$ G3$2$ )-%*& +*&*'$)%$ ;3$ $.%*.(2$)%(2*2$. .-) 94+(2*./ 0* ;3$ $.%*. &*8-)$. .-) .('G+$'$)%$ +-. &$ NG&- -. 2$ +*. &*8-L)$. =4.( *./ +- ;3$ J* $ ;3$ 3.3*+'$)%$ )- *G*&$8 *) %$ +*. .$G*&*2*. G*&* $++*. $) +*. *+ 3+*2-&*.6 O(.%H&( *'$)%$/ .3. 9*+-&$. *G*&$ N*) $) +*. %*=+*. 2$ %&(1-)-'$%&N*/ 0 .$&L9N*) G&$ (.*'$)%$ G*&* -=%$)$& +-. 9*+-&$. 2$ +-. &$ NG&- -. 2$ +*. M3) (-)$. %&(1-)-'I%&( *.=4.( *.6
!"!" #$%&'() *#+,&'&-.*#+/$) !"#$%&' ()*+$()&)*' ,-+)&.*+$&/*' '0+ /&' '(1-(*+$*'tan(α) =
sen(α)
cos(α), cot(α) =
cos(α)
sen(α) !"#$%&' )*+, 20.0 sen(α) =a
c3 cos(α) =
b
c4 $*+*.0' 5-*
tan(α) =a
b=
a
cb
c
=sen(α)
cos(α).6* /& .('.& ,0%.& '* )*.-*'$%& 5-* cot(α) =
cos(α)sen(α)7& '(1-(*+$* ()*+$()&) '* 0+0 * 0.0 7& 9)*+$()&) :($&1;%( & 3 )( * 5-*
cos2(α) + sen2(α) = 1 !"#$%&' )*+, <*& α */ =+1-/0 )* -+ $%(=+1-/0 %* $=+1-/0 0>-*'$0 &/ &$*$0 a 3 '*& c /&?(>0$*+-'&@ A+$0+ *'4 >0% */ $*0%*.& )* :($=10%&' $*+*.0' 5-*a2 + b2 = c2
=⇒ a2
c2+
b2
c2= 1
=⇒ (a
c)2 + (
b
c)2 = 1
=⇒ sen2(α) + cos2(α) = 1.
B 0+$(+-& (;+ C*%*.0' 0$%&' )0' ()*+$()&)*' ,-+)&.*+$&/*'1 + tan2(α) = sec2(α), 1 + cot2(α) = csc2(α)
!" !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+ !"#$%&' )*+,1 + tan2(α) = 1 +
sen2(α)
cos2(α)
=cos2(α) + sen2(α)
cos2(α)
=1
cos2(α)
= csc2(α)#$% & (& )*+,-)*$*+% ./,*$0+,-$1+% 2/+ (+0&% +%-/*)$*& +, +%-$ %+ )3, %+ *+4+, 0+0&5)6$57$ 2/+ ,&% %+58, ),*)%9+,%$41+% 9$5$ :+5); $5 0/ ($% &-5$% )*+,-)*$*+%< 7 $*+08% 9$5$%)091); $5 +=95+%)&,+% -5)>&,&0?-5) $% &091) $*$%@ A1 -5$4$B& &, )*+,-)*$*+% +% C-)1 9$5$),-+5,$1)6$5 1$% 5+1$ )&,+% 48%) $% 7 *+%$55&11$5 +1 5+ &,& )0)+,-& *+ 9$-5&,+%@ !"# % %&' ()*)* ! "#$# #&# '() &* +), -(.'+), #.'&), &* +# /.'$# 0#++#$ ,' ),*()1 ,*() 2 3#(.*(3*!.........................................................................................................................................
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................................................................................................................................................................ ..................
(a)
7
3
......................................................................................................................................................................................................................................................................................................................................................................................... .
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..........
(b)
12
15
......................................................................................................................................................................................................................................................................................................................................................................................... .
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..........
(c)√3
54! "#$# #&# '() &* +), -(.'+), 5(&5 #&),1 0#++#$ ,' ),*()1 ,*() 2 3#(.*(3*!................................................................................................................................................................................................................................................................................
..............
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······················································································
α
····································································
······························ β································
································································
········································
γ ········································································································································
δ6! 7#++#$ *+ ),*() 2 +# 3#(.*(3* &*+ -(.'+) #.'&) '2) ,*() *, 5.'#+ #8#9 :; 8<9 1/√3 8 9 5/13 8&9 5/3;! 7#++#$ *+ ,*() 2 +# 3#(.*(3* &*+ -(.'+) #.'&) '2) ),*() ='*$# 5.'#+ #8#9 :; 8<9 >: 4 8 9 1/
√3 8&9 >:6
!"!" #$%&'() *#+,&'&-.*#+/$) !! ! "#$$#% &$ '&() * &$ )'&() ,&$ -(./$) #./,) /*# 0#(.&(0& 1/&%# 2./#$ #3#4 567 384 9:6 3 4 √3/3 3,4 7/12;! 3#4 "#$$#% &$ '&() * $# 0#(.&(0& ,& α< /#(,) cos(α) = x!384 "#$$#% &$ )'&() * $# 0#(.&(0& ,& α< /#(,) sen(α) = x!3 4 "#$$#% &$ )'&() * &$ '&() ,& α< /#(,) tan(α) = x!=! "#$$#% $# >&,2,# ,& $)' #0&0)' * $# ?2@)0&(/'# ,& #,# /() ,& $)' '2./2&(0&' 0%2-(./$)'%& 0-(./$)'!
......................................................................................................................................................................................................................................................................................................................................................................................... ............
(a)
7
27◦
......................................................................................................................................................................................................................................................................................................................................................................................... ............
(b)15
35◦
......................................................................................................................................................................................................................................................................................................................................................................................... ............
(c)√3
65◦A! "#$$#% &$ -%&# ,& /( 0%2-(./$) %& 0-(./$)< '#82&(,) B/&3#4 '/' #0&0)' >2,&( ! * 5! >! %&'@& 02C#>&(0&!384 /() ,& '/' -(./$)' #./,)' >2,& 5D◦ * &$ $#,) )@/&'0) >2,& :D >!3 4 /() ,& '/' -(./$)' >2,& ;◦ * &$ $#,) #,*# &(0& >2,& := >!3,4 /() ,& '/' -(./$)' >2,& ;D◦ * $# #$0/%# )%%&'@)(,2&(0& #$ -(./$) %& 0) >2,& 9D >!E! "#$$#% &$ -%&# ,&$ '2./2&(0& 0%2-(./$) %& 0-(./$)
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
α β
γ
A B
C
ab
c'#82&(,) B/&3#4 c = 50 * α = 38◦! 384 a = 45 * β = 65.5◦!9D! "#$$#% $# >&,2,# ,&$ @&%F>&0%) ,& /( 0%2-(./$) %& 0-(./$) △(ABC) '#82&(,) B/&3#4 a = 30 * ∠C = 80◦! 384 c = 50 * β = 35.5◦!99! G( @)'0& @%)*& 0# &( 2&%0) >)>&(0) ,&$ ,F# /(# ')>8%# ,& :5! >&0%)'! H2 &$ -(./$) B/&1)%># $# $F(&# ,& C2'2I( ,& $# @/(0# ,&$ @)'0& %&'@& 0) #$ '/&$) ,&',& $# @/(0# ,& $# ')>8%# &',& 55◦< J /-$ &' $# #$0/%# ,&$ @)'0&K9:! H& ,&'&# #>#%%#% /( @)'0& C&%02 #$ ,& #$0/%# ,& 5 @2&'< )( #8$&' B/& 2%-( ,&',& $# @/(0# ,&$@)'0& #$ '/&$)! JL/M $)(.20/, 0&(,%F# #,# #8$&< '2 &$ -(./$) B/& 1)%>#%- )( &$ '/&$) '&%- ,&=: ◦K
!" !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+ !" #$%&'$ ()'*+% (* '$,(-.*/% ,.%-0/$1 % 3(. )1 α 4 β )%* 5*,(6%) %-76.-.*/'$1%) .*/%* .)cos(β) = sen(α) 4 sen(β) = cos(α).8('*+% *% .91)/:'* '6 (6'+%$'); ). 7$.7'$'&'* /'&6') %*/.*1.*+% 6%) <'6%$.) +. %).*% 4).*%; 7.$% )%6'-.*/. 7'$' 5*,(6%) θ /'6.) 3(. 0◦ ≤ θ ≤ 45◦" =9761 '$ 7%$ 3(0 .)/') /'&6')).$<:'* 7'$' '6 (6'$ 6%) <'6%$.) +. ).*% 4 %).*% 7'$' ('63(1.$ 5*,(6% ',(+%" >" 8%-76./'$ 6' )1,(1.*/. /'&6'; )1,(1.*+% .6 7'/$?* 1*+1 '+% .* 6' 7$1-.$' @6'; +%*+. '+' %6(-*' .97$.)' 6' $'A?* +.6 .* '&.A'+% +. 6' %6(-*' .* /0$-1*%) +. 6' $'A?* 1*+1 '+' '61*1 1% +. 6' @6'" !" .) (*' $'A?* /$1,%*%-0/$1 ' +.@*1+' %-% .6 $. :7$% % +. /'*,.*/."
sen(α) cos(α) tan(α) cot(α)
sen(α) sen(α)√
1− sen2(α)sen(α)
√
1− sen2(α)
√
1− sen2(α)
sen(α)
cos(α)tan(α)cot(α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
!"#" $%&'()&*+ ,-(.)&)/0,-('1+ !"#$%&' (') #*+&, (' -+&'%.' -0+ (' )$ -% *+1'%'+ -$ ,+ ') .'2-'3' #,.-&-4, (' X5 '. (' -%5 ')#*+&, ,+ ,,%('+$($. (R, 0)5 (,+(' R '. ') %$(-, (' )$ -% *+1'%'+ -$ 6*' 2,(')$ )$ #-.&$78'% )$ .-9*-'+&' :9*%$ (,+(' ') #*+&, M %'#%'.'+&$ $) 2,&,%-.&$7..............................................................................................................................................................................................................................................................................................
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.......................................................................................
...........................................................................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................• •
•
A
M
O !! ";-9*%$ !<7=> ?,4-2-'+&, -% *)$% !"#$%& '()*)') @*#,+9$2,. 6*' )$ #-.&$ -% *)$% &-'+' !AA 2'&%,. (' %$(-, B 6*' ')2,&,%-.&$ .' 2*'4' ,+ *+$ %$#-('C (' = 2D.'9 '+ )$ (-%' -0+ #,.-&-4$5 #$%&-'+(, (' A =
(100, 0)7!7 EF*G) '. ') )$%9, (' )$ #-.&$HI.$%'2,. )$ 10%2*)$ L = 2πR #$%$ ,2#*&$% (- J, )$%9,7 K'+'2,.5 '+&,+ '.5 6*'L = 2πR = 200π ≈ 628m7<7 EF*G+&, &-'2#, )' &,2$%G $) $%%, ($% *+$ 4*')&$H L .'$5 E *G) .'%G .* #'%M,(,H#'%M,(, =
d
v=
200π
5= 40π ≈ 125.7 .'97"7 EF*G+&, &-'2#, )' &,2$%G $) $%%, ($% 2'(-$ 4*')&$HN$ 2-&$( (') &-'2#, #$%$ ($% *+$ 4*')&$5 6*' '. 20π ≈ 62.85 .'97 7 EF*G+&, &-'2#, )' &,2$%G ($% (-'C 4*')&$.HO-'C 4' '. ), 6*' )' &,2$ ($% *+$ 4*')&$5 6*' '. 400π ≈ 1257P+ ),. 2,4-2-'+&,. -% *)$%'.5 ))$2$2,. +," ."/ 01 $) +Q2'%, (' 4*')&$. , - ),. ,2R#)'&$(,. '+ *+$ *+-($( (' &-'2#,7 P) $",2&3& '. ') &-'2#, '2#)'$(, '+ ,2#)'&$% *+$4*')&$7P+ ') '3'2#), $+&'%-,% ') 204-) &'+M$ *+ #'%M,(, (' 40π .'97 P+ ,&%$. #$)$S%$.5 )' &,2$S$
40π .'9*+(,. ($% *+$ 4*')&$7 N)$2$+(, f $ .* 1%' *'+ -$5 &'+'2,. '+&,+ '. 6*'f 4*')&$.1 .'9*+(, =
1 4*')&$40π .'9*+(,. .
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...............................................................................................................................................
............................................................................................................................................................................................................................................................................................................ • •
•
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t=0
t=1
t=2
t=3
t=5
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t=6 Q3?)., !7$AR S6<3530'/6 3. )4,.8)'/6 (0, 5) H%86. ()*9I$ T4,50561 P = P (t) , 4, 8613 3E' -04 5E<34 4)0?6 -0 /.,'1 )..3-6t 10?)'-61 C 1)86'?,561 ()0 P (t) = (x(t), y(t))$ U 8,./3. -04 /30586 /.,'1 )..3-6 86-0561-0 3. ,4?)',1 61,1 , 0. , -0 4, 8613 3E'$ &' 8,./3 )4,.= 0' 01/0 0G05846 .01)4/,.; E56-6-01 .323. 04 ;'?)46 θ 656 D)' 3E' -04 /30586$ &' 0D0 /6= ),'-6 t = 1= 04 5E<34 :,2.;.0 6..3-6 !V!7 -0 <)04/,= ()0 0()3<,40= 0' /*.53'61 -0 ?.,-61= , -0 3. ()0 P 01/; 0' )',8613 3E' /,4 ()0 θ = 360/12 = 30◦$ T, /,24, 13?)30'/0 5)01/., 04 ;'?)46 θ 0' 461 8.350.6110?)'-61= 65840/,. 4,1 64)5',1 <, K,1$/30586 0.0 1.0 1.5 2.0 2.5 3.0 4.0 5.0 6.0
θ 0◦ 30◦ 45◦ 90◦ 180◦
!" !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+#$ %& '$(%)*+&, -% ('%./, 0 ≤ t ≤ 6 /,-%.,0 ,1(%$%)2 ,$ &+ +45-+ -% &+ ()'6,$,.%()7+25$+ -%0 )'/ '8$ -% &+0 95$ ',$%0 x(t)2 y(t): ;%+ P = (x(t), y(t)) 4 ,$0'-%)%.,0 %& <$65&, θ:=,., 0+1%.,0 >5% %& )+-', %0 ?2 %$ 5$ '$0(+$(% t (%$%.,0 >5% @*%) &+ 0'65'%$(% A65)+Bx(t) = 5 cos(θ), y(t) = 5 sen(θ).
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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A=(5,0)0
P (t)=(x(t),y(t))
x(t) = 5 cos(∠(AOP ))
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r 19( )+) /, 1, %(%7/.) +/ &&*+/,)+)% +/.),/*) 7)0 :1/ %1 /,7*& &(, (+/ &, /0 &*(3/, O +/0 %(%7/.)# 6&,%(+/*).&%5 /, /0 .(%.&%(%7/.) +/ &&*+/,)+)%5 1, 2,310& /, =&%( (&, /%72,+)*# B&7/ :1/ /0 0)+& (,( ()0 +/0 2,310& &*7) 0) (* 1,A/*/, () /, /0 =1,7& (r, 0)5 .(/,7*)% :1/ /0 0)+& 7/*.(,)0 +/0 2,310&5 &*7)0) (* 1,A/*/, () /, 1, =1,7& :1/ %(.9&0(C)*/.&% =&* P # B&7/ :1/ /0 =1,7& P =1/+/ /%7)*)@&*) /, 1)0:1(/* 1)+*),7/ & />/ +/=/,+(/,+& +/0 7).)D& +/0 2,310&#........................................................................................................................................................................................................................................................................................................
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............................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................
A
O
B
P
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..............................................................................................................E(31*) !F#GH I,310&% /, /0 =0),&J&* &7*& 0)+&5 %( =*(./*& /% &3/.&% 1, =1,7& +/ 0) (* 1,A/*/, ()5 +(3).&% B5 7/,/.&%+/K,(+& 1, 2,310&5 ∠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α /,7*/ O 4 PQO 3*)+&%# 6&.& /%7) &**/%=&,+/, () /% -,( )5 =&+/.&% +/K,(* /%7)%*)C&,/% &.& A1, (&,/% :1/ 00).).&% (&! )*$ ,-$ ! "',.-$-/0"', &! % α#
!" !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+ !"#$ $&# '! ()# $*#!+ ,-$.*#*/01-$ 2+ 32-2 )# 4#.)5* α !# !5352#*#$% α &' (')&*+ $' ,+-. .0' $-1('2%3 $' $* ,*%'+4 1%* 5&$ -& *%2+ 1$36.'%* +31% *% .3 &'7$3$' .%4 +' $'13+ (0, 0) 8 3%2.+ r4 $' $* ,&'1+ P = (xP , yP )9:'1+' $-4cos(α) :=
xPr, sen(α) :=
yPr67+!-82 $*#!+9!9 ;<-$3=$6+-4 $' ,3.6$3 *&)%34 5&$ '+ >%8 +'7&-.0' +' *% '+1% .0' %'1$3.+3 2$ *+ 8+!#4 8% 5&$ *%- '&$=%- '+ .+'$- 8 *%- %'1$3.+3$-4 +.' .2$' ,%3% (')&*+- %)&2+-9?9 @01$-$ %2$6(- 5&$ ,%3% &' (')&*+ α +' ,&'1+ 1$36.'%* P 4 *% ,%3$A% 2$ 'B6$3+-
cos(α)4 sen(α) 1.$'$' -.)'+- 5&$ +33$-,+'2$' +' *+- -.)'+- 2$ *%- ++32$'%2%- 2$*,&'1+ P 9C9 #. *% .3 &'7$3$' .% &-%2% $' *% 2$D'. .0' $- &'% .3 &'7$3$' .% &'.1%3.%4 $- 2$ .34 -. $*3%2.+ 7&$-$ !4 $'1+' $- $* +-$'+ 8 $* -$'+ 2$* (')&*+ -$3E%' *%- ++32$'%2%- 2$* ,&'1+P 4 2$ 6%'$3% 5&$
cos(α) =xp1
= xp, sen(α) =yp1
= yp 9 :' $* ,*%'+4 ,+2$6+- +'-.2$3%3 (')&*+- ,+-.1.=+- 8 '$)%1.=+-9 F&%'2+ 3+1%6+- $*-$6.$A$ ,+-.1.=+ 2$ x $' +'13% 2$ *%- 6%'$ .**%- 2$* 3$*+A4 2$ .6+- 5&$ $* (')&*+7+36%2+ $- ,+-.1.=+9 #. ).3%6+- % 7%=+3 2$ *%- 6%'$ .**%- 2$* 3$*+A4 2$ .6+- 5&$ $*(')&*+ $- '$)%1.=+9G9 :' $* ,*%'+ ,+2$6+- 1$'$3 (')&*+- 5&$ 2$' 6(- 2$ &'% =&$*1%4 8 $' $-1$ %-+ $* (')&*+,&$2$ 6$2.3 6(- 2$ 360◦ + 2π 3%2.%'$-9H9 F&%'2+ $* *%2+ 1$36.'%* 2$* (')&*+ +.' .2$ +' &'+ 2$ *+- $A$- 2$* ,*%'+4 2$ .6+- 5&$$- &' (')&*+ )2'-2#1259I9 :* &%23%'1$ 2$ &' (')&*+ $- %5&$* $' $* &%* 2$- %'-% -& *%2+ 1$36.'%*9 J$ 6%'$3%5&$4 2$ .6+- 5&$ &' (')&*+ 2$ 120◦ $-1( $' $* -$)&'2+ &%23%'1$ 8 &' (')&*+ 2$ 235◦$-1( $' $* 1$3 $3 &%23%'1$9"9 F&%'2+ 2+- (')&*+- +6,%31$' $* 6.-6+ *%2+ 1$36.'%*4 2$ .6+- 5&$ -+' *1!-/$#2:5!+9 K+2$6+- >%**%3 (')&*+- +1$36.'%*$-4 ,+-.1.=+- + '$)%1.=+-4 -&6%'2+ + 3$-1%'2+
!"#" $%&'()&*+ ,-(.)&)/0,-('1+ !"#$%&'() +,-&%'() (& ./0$&+ 1(1+2 3+4 %5%,-&+6 $/ ./0$&+ 1% 60◦ %) +'%4,7/(& +/ $/./0$&+ 1% 420◦ 8 +/ $/ ./0$&+ /%0('7#+ 1% −300◦"2 9+) ./0$&+) +'%4,7/(&%) '7%/%/ &() ,7),() :$/ 7+/%) '470+/+,%'47 () 8( ;$% +,-('%/%& -$/'+ P 2<+/+ 71() &() :$/ 7+/%) '470+/+,='4 () -(4( &+) ./0$&+) %)-% 7(&%) %/ %& -47,%4 $(14(/'%6%& >,-$'+ 1% &() :$/ 7+/%) -(4( ./0$&+) %/ %& )%0$/1+6 '%4 %4 8 $(4'+ $(14(/'%) )% -$%1%/+?'%/%4 ( -(4'74 1% %)'()2 !"#$%& '! '$ &'"#!(% #*(+*!,'<$(&;$7%4 ./0$&+ 1%& )%0$/1+ $(14(/'%6 -$%@1% %A-4%)(4)% +,+ 180◦ − α6 1+/1% α %) $/./0$&+ (0$1+2 B74(/1+ &( C0$4(6 #%,+) &() )7@0$7%/'%) 4%&( 7+/%) 0%+,='47 ()OB′ = OC ′ 8 BB′ = CC ′.D )%( ;$% )% '7%/%/ &() )70$7%/'%) 4%&( 7+/%)
cos(180◦ − α) = − cos(α)sen(180◦ − α) = sen(α).
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α
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O A
B
C
◦ ◦ ◦
◦
◦
B′
C′
!"#$%& '! '$ ,'+ '+ #*(+*!,'<$(&;$7%4 ./0$&+ 1%& '%4 %4 $(14(/'%6 -$%1%%A-4%)(4)% +,+ 180◦ + α6 1+/1% α %) $/ ./@0$&+ (0$1+2 B74(/1+ &( C0$4(6 #%,+) &() )7@0$7%/'%) 4%&( 7+/%) 0%+,='47 ()OB′ = OC ′ 8 BB′ = CC ′.D )%( ;$% )% '7%/%/ &() )70$7%/'%) 4%&( 7+/%)
cos(180◦ + α) = − cos(α)sen(180◦ + α) = − sen(α).
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O A
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!" !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+ !"#$%& '! '$ #)*+% #),*)!+'#$%&'$()* +,-$&. /)& $%*1. $%/*%,1)2 3$)4/) )53*)6%*6) .7. 360◦ − α2 /.,/) α )6 $,+,-$&. %-$/.8 9(*%,/. &% :-$*%2 ;)7.6 &%6 6(4-$(),1)6 *)&% (.,)6 -).7<1*( %6BB′ = CB′.= 6)% '$) 6) 1(),), &%6 6(-$(),1)6 *)&% (.,)6
cos(360◦ − α) = cos(α)sen(360◦ − α) = − sen(α)
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O A
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-.'/0$%& 1234353 >$3.,(),/. .,. (/.6 &.6 ;%&.*)6 /) .6),. ? 6),. 3%*% +,-$&.6 %-$/.6)63) (%&)62 @%&&) &. 6(-$(),1)8A8 cos(150◦) = cos(180◦−30◦) = − cos(30◦) = −√3/28 B-$%&7),1)2 sen(150◦) = sen(30◦) =
1/28!8 cos(240◦) = cos(180◦+60◦) = − cos(60◦) = −1/28 B-$%&7),1)2 sen(240◦) = − sen(60◦) =−√3/28C8 cos(300◦) = cos(360◦ − 60◦) = cos(60◦) = 1/28 B-$%&7),1)2 sen(300◦) = − sen(60◦) =
−√3/28 !"#$%& 6#),*)!+)$'&D) .*/%7.6 '$) &.6 +,-$&.6 $%/*%,1%&)6 6., %'$)&&.6 $?. &%/. 1)*7(,%& .(, (/) ., $,./) &.6 )E)6 /)& 3&%,.8 F#$+&)6 6., &%6 G$, (.,)6 1*(-.,.7<1*( %6 /) )61.6 +,-$&.6H >( *) $4**(7.6 % &% (* $,G)*), (% $,(1%*(%2 ,.1%*)7.6 '$) &.6 3$,1.6 /.,/) )61.6 +,-$&.6 .*1%, &% (* $,G)*), (% 6., IA2"J2 I"2AJ2 I4A2"J ? I"24AJ8 K.* &. 1%,1.2 %&-$,.6 )E)73&.6 /) &%6 G$, (.,)61*(-.,.7<1*( %6 /) )61.6 +,-$&.6 6.,
sen(180◦) = 0, cos(180◦) = −1, tan(180◦) =sen(180◦)
cos(180◦)=
0
1= 0=L6)*;) '$)2 6( .,6(/)*%7.6 1./.6 &.6 +,-$&.6 ), )& 3&%,.2 ? ., )61% ,$);% /):,( (M,21),)7.6 '$) ),1*) " ? 360◦ )5(61), /.6 +,-$&.6 '$) 1(),), )& 7(67. .6),. ? /.6 +,-$&.6 .,)& 7(67. 6),.2 )5 )31. 3%*% &.6 +,-$&.6 $%/*%,1%&)68 N) 7%,)*% '$) sen(α) = sen(180−α)? '$) cos(α) = cos(360 − α)
!"#" $%&'()&*+ ,-(.)&)/0,-('1+ !" !"#$%& '()*)+) #$%%$& '()(* %(* +,-.%(* α '$%/* 0./ sen(α) = 1/212$ *(%. 45, 6&4, 46$% /* arc sen(1/2) = 30◦7 6/&( '/,/8(* .,$ *(%. 45, $)4 4(,$%7 180◦ −300◦ = 150◦1 !"#$%& '()*),) #$%%$& '()(* %(* +,-.%(* '$%/* 0./ cos(α) = −1/212$ *(%. 45, 6&4, 46$% /* arc cos(−1/2) = 120◦ 9(:'/,4)$ )/ 8/8(&4$ ( (, %$ $;.)$ )/ .,$ $% .%$)(&$1 <(& %$* &/%$ 4(,/* /*'.)4$)$* '/,/8(* 0./ 360◦−120◦ = 240◦ /* %$ ('&$ *(%. 45,1 !"- / /&0 '()*) ! "#$%&'( '*' %+, *- .,/ /#0%#-+1-/ 2+0%.,/ -+ 3,/# #4+ -/12+*'(!5'6 30◦! 5$6 45◦! 5 6 150◦!5*6 225◦! 5-6 300◦! 576 325◦!8! 9"- .'/ 8 *- .' :-*#'+, ;- ' .'/ < *- .' :'='+'> ?%@ 2+0%., (- ,((- -. ;,('(#, 5.' '0%&' *-.'/ ;,('/6 *- %+ (-.,& :- 2+# ,AB! C'..'( -. 2(-' *- %+ /- 1,( '+0%.'( *- %+ D( %., *- ('*#, E %F, 2+0%., -+1('. :#*- 8GH◦!G! I' !""# $%&'! # /- *-J+- ,:, .' .,+0#1%* *-. '( , /%$1-+*#*, -+ .' /%3-(J #- *- .' 1#-(('3,( %+ 2+0%., -+1('. *- :#+%1,! K%3,+#-+*, ?%- -. *#2:-1(, *- .' 1#-((' -/ LM8L :#..'/>;'..'( .' '+1#*'* *- :#..'/ -+ %+' :#..' +2%1# '!E! K%3,+0' ?%- α -/ %+ 2+0%., '0%*, 1'. ?%- cos(α) = 0.6! C'..'( -. N'.,( *- sen(α)> tan(α F
cot(α) F ,:3.-1'( .' 1'$.' /#0%#-+1-> /#+ %/'( '. %.'*,('!x α 180− α 180 + α 360− α
cos(x) .6sen(x)tan(x)cot(x)O! P/'+*, .'/ (-.' #,+-/ -/1%*#'*'/QF .,/ N'.,(-/ ,+, #*,/ 3'(' .,/ 2+0%.,/ *- BH◦> GE◦F OH◦Q ,:3.-1'( .' /#0%#-+1- 1'$.'! R, %/- '. %.'*,('> 3-(, /# %+ *#$%&, '*- %'*,!210◦ 225◦ 240◦ 270◦ 300◦ 330◦ 360◦
cossentanL! C'..'( .'/ ,,(*-+'*'/ *-. 3%+1, 1-(:#+'. /,$(- %+' #( %+7-(-+ #' %+#1'(#' ,+ -+1(, -+ -.,(#0-+> *- 2+0%.,/ ,+ :-*#*'/ 8 H◦> 88E◦F BBH◦-+ 3,/# #4+ -/12+*'(
!! !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+ ! "#$$#% $#& ((%)*+#)#& )*$ ,-+.( .*%/0+#$ &(1%* -+# 0% -+2*%*+ 0# )* %#)0( 3 (+ *+.%( *+*$ (%04*+5 )* 6+4-$(& (+ /*)0)#& 789◦5 773◦: ;;3◦!<! =%(1#% $#& &04-0*+.*& %*$# 0(+*&5 )(+)* α *& -+ 6+4-$( #4-)(!>#? tan(180◦ + α) = tan(α)! >1? cot(180◦ + α) = cot(α)!> ? tan(360◦ − α) = − tan(α)! >)? cot(360◦ − α) = − cot(α)!89! @*# a -+ +A/*%( .#$ B-* −1 < a < 0>#? C(&.%#% 4%6D #/*+.* B-* $# * -# 0E+ sen(x) = a .0*+* *F# .#/*+.* )(& &($- 0(+*&5 -+#*+ *$ -#%.( -#)%#+.* : $# (.%# *+ *$ .*% *% -#)%#+.*!>1? C(&.%#% 4%6D #/*+.* B-* $# * -# 0E+ cos(x) = a .0*+* *F# .#/*+.* )(& &($- 0(+*&5 -+#*+ *$ &*4-+)( -#)%#+.* : $# (.%# *+ *$ .*% *% -#)%#+.*!88! G&#+)( -+# #$ -$#)(%#5 H#$$* *$ I#$(% )* arc sen(sen(210◦))! JF,$0B-* *$ I#$(% (1.*+0)(5 *&,*K 0#$/*+.* *$ &04+( +*4#.0I( )* $# %*&,-*&.#!87! "#$$#% (&*+(5 &*+( : .#+4*+.* )* $(& 6+4-$(& *+ ,(&0 0E+ *&.6+)#% -:( $#)( .*%/0+#$ (+.0*+*#$ ,-+.( B-* &* 0+)0 #!>#? (5, 5)! >1? (6,−8)! > ? (5,−12)!>)? (−2, 3)! >*? (−2,−3)! >2? (√3,√2)!8;! L+)0 #% *$ -#)%#+.* )(+)* &* H#$$# *$ 6+4-$( x .#$ B-*>#? cos(x) > 0 : sen(x) < 0! >1? tan(x) > 0 : sen(x) < 0!> ? cot(x) > 0 : sen(x) > 0! >)? sen(x) < 0 : cos(x) < 0!>*? tan(x) > 0 : cos(x) > 0! >2? cot(x) > 0 : cos(x) < 0!8M! "#$$#% .()(& $(& 6+4-$(& α (+ /*)0)#& *+ *$ 0+.*%I#$( [0, 360) .#$*& B-*>#? sen(α) = −1/2! >1? cos(α) = −1/2! > ? tan(α) = 1!>)? tan(α) = −1! >*? cos(α) = 1! >2? sen(α) = −1!83! @*# a -+ +A/*%( .#$ B-* 0 < a < 1>#? C(&.%#% 4%6D #/*+.* B-* $# * -# 0E+ sen(x) = a .0*+* *F# .#/*+.* )(& &($- 0(+*&5 -+#*+ *$ ,%0/*% -#)%#+.* : $# (.%# *+ *$ &*4-+)( -#)%#+.*!>1? C(&.%#% 4%6D #/*+.* B-* $# * -# 0E+ cos(x) = a .0*+* *F# .#/*+.* )(& &($- 0(+*&5 -+#*+ *$ ,%0/*% -#)%#+.* : $# (.%# *+ *$ -#%.( -#)%#+.*!8N! "#$$#% .()(& $(& 6+4-$(& α (+ /*)0)#& *+ *$ 0+.*%I#$( [0, 360) .#$*& B-*>#? sen(α) = −1/2! >1? cos(α) = −1/2! > ? tan(α) = 1!>)? tan(α) = −1! >*? cos(α) = 1! >2? sen(α) = −1!8O! @*# a -+ +A/*%( .#$ B-* 0 < a < 1
!"#" $% &'()(% (' $*+ %,-*+. '$ ,%()/0 !" !" #$%&'!' (')* !,-.&- /0- 1! - 0! 23. sen(x) = a &2-.- -4! &!,-.&- 5$% %$10 2$.-%6 0.!-. -1 7'2,-' 0!5'!.&- 8 1! $&'! -. -1 %-(0.5$ 0!5'!.&-9 :" #$%&'!' (')* !,-.&- /0- 1! - 0! 23. cos(x) = a &2-.- -4! &!,-.&- 5$% %$10 2$.-%6 0.!-. -1 7'2,-' 0!5'!.&- 8 1! $&'! -. -1 0!'&$ 0!5'!.&-9 " ;!11!' <3',01!% 7!'! cos(270− α) 8 sen(270− α) -. &=',2.$% 5- cos(α) 8 sen(α)9>?9 ;!11!' <3',01!% 7!'! cos(270 + α) 8 sen(270 + α) -. &=',2.$% 5- cos(α) 8 sen(α)9>@9 A$. !.&-'2$'25!5 ! 1! 2.B-. 23. 5- 1!% !1 01!5$'!% -1- &'3.2 !% 1! ,!8$'C! 5- 1$% B!1$'-% 5- 1!%'!D$.-% &'2($.$,=&'2 !% %- $:&-.C!. 5- &!:1!% 5- B!1$'-%9 E2 F!% &!:1!% $.&-.C!. 1$% B!1$'-%5- %-.$6 $%-.$6 &!.(-.&- 8 $&!.(-.&- 7!'! ).(01$% x &!1-% /0- 0◦ ≤ x ≤ 45◦9 GA3,$ %-7$5'C! 0%!' -%! &!:1! 7!'! 5-&-',2.!' cos(67◦)6 sen(140◦H !"#" $% &'()(% (' *+, -. +,0 '* 1%()23#$%&'()*)+$& ,% +$-$*'&-. /'.0.%($ &$1*) ,%. 2'&-. '* ,4.* $% *.2'()5 $%&-.%-)6 7,82$%9.+$& :,) 4$& )0)& () $$*()%.(. )&-;% $4$ .($& () 4. <$*+. ,&,.4 = .()+;& :,) )42,%-$ () 2.*-'(. >($%() )4 -')+2$ t = 0? )& A6 @'%.4+)%-)A ()%$-)+$& 2$* P 4. 2$&' 'B%()4 +$-$*'&-.6 C +)('(. :,) )4 +$-$*'&-. &) +,)/)A '+.9'%)+$& :,) -.+1'D% &) +,)/) )4&)9+)%-$ OP =A 2$* 4$ -.%-$A )4 ;%9,4$ α )&-; .+1'.%($6 E) '+$& :,) 4. *.2'()5 () .+1'$() )&) ;%9,4$ )& 4. !"# %&'& '()*"'+ ()4 +$-$*'&-.6F$* )0)+24$A &,2$%9.+$& :,) )4 +$-$*'&-. (. ,%. /,)48-. )% ,.-*$ +'%,-$&6 #$+$ )&-$ ):,'/.4) . ,%. *$-.8 'B% ()4 &)9+)%-$ OP () "GH◦)% +'%,-$&A -)%)+$&,%. /)4$ '(.( .%9,4.* ()360
4= 90
9*.($&+'%,-$& . ....................................................................................................................................................................................................................................................
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x
1000=
2πR
360=⇒ x = 1000 · 12π
360=
1000π
30≈ 104.7 91
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75=
π
180=⇒ x =
75
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x
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sen(−t) = − sen(t).5"10#/,+ ()%! H3* t 6/ /B'3%$ %3*2+ 3/,$/ 30 2$0 5/162$0 *0)1/*($0 * t @ (−t) 0$/ 0)'.,%) $0%3043 ,$ *2 383 X7 H) 32 4%)'3%$ 30 (x, y) 32 0316/($ 03%5 (x,−y)+ (3 ($/(3 03 $-,)3/3/ 2$0%3062,*($07E$ */,3%)$% )/() * =63 2* 1%5: * (3 $03/$ 30 0)'.,%) * %3043 ,$ *2 383 Y + ')3/,%*0 =63 2*1%5: * (3 03/$ 2$ 30 %3043 ,$ *2 $%)13/7 !"#"!" 6,78 )* 0/ 1'( .-(/* 9,.&-(-453,. )*#$% 2* 43%)$() )(*( (3 30*0 ;6/ )$/30+ -*0,*%5 $/ $-,3/3% 2*0 1%5: *0 4*%* D*2$%30 (3 x ,*230=63 0 ≤ x ≤ 2π+ @* =63 2$ (3'50 03%5 6/* %343,) )</ 43%)<() * (3 30,* 4*%,37A$'3/>*%3'$0+ 4$% 2$ ,*/,$+ $-,3/)3/($ 2$0 D*2$%30 (3 $03/$ @ 03/$ 4*%* D*2$%30 (3 x 3/,%30 @ 2π7
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cos(t)
sen(t)#$%&*+*+ /2/ +(/2')*-$/ 2/*3-$ 23* *' 2'*-$+*7 8*/ *' 2'*-$+*/ &2(-(3 $%&2)*+ -06+( )*%(3)( '$/ .*'$+(/ -( $/(3$ 9 /(3$ 2*3-$ '$/ :312'$/ (/):3 (;&+(/*-$/ (3 +*-0*3(/4<=>? (/ 3( (/*+0$ *@2/)*+ '* %$-*4 23* A23 0$3*'0-*- B2( -(&(3-( -( /2 %:B203*4 &(+$ B2(&(+%0)( 03A$+%*+'( * '* %:B203* B2( (/)*%$/ +(*'0C*3-$ '$/ D%&2)$/ (3 +*-0*3(/7 =3 '$/'(312*@(/ -( &+$1+*%* 0D3 -( $%&2)*-$+*/4 2/2*'%(3)( E*9 &+(-(F30-*/ A23 0$3(/ $%$/(34 $/4 )*3 9 /( /2&$3( B2( '$/ *+12%(3)$/ (/):3 (3 +*-0*3(/7.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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x
y = cos(x)
π6
π3
2π3
5π6
7π6
4π3
3π2
5π3
11π6
π 2ππ2
1 ..................................................................................................
.....................................................................................................................................
..
..................................................................
............................................................................................................................................................................................................................................................................................• •
••
•
••• • •
•
•
•• •
G012+* "H7"HI J+:F * ,:/0 * -( #$/(3$.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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x
y = sen(x)
1
π6
π3
2π3
5π6
7π6
4π3
3π2
5π3
11π6
π 2π
π2
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....................................................................................................................................................................
.........................................
.............................................................................................................................................................................................................
........................................................................................................................................•
•• • •
•
•
•• • •
•
•
G012+* "H7"!I J+:F * ,:/0 * -( '* A23 0D3 K(3$L/*3-$ '$/ .*'$+(/ $3$ 0-$/ -( $/(3$ 9 /(3$ )(3(%$/ '*/ 1+:F */ &*+* (/*/ A23 0$3(/4 B2((/):3 %$/)+*-*/ (3 '*/ F12+*/ "H7"H 9 "H7"!7
!" !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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x
y = sen(x)
• • • • • • • •
1
−6π −4π −2π 2π 4π 6π 8π......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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x
y=cos(x)
• • • • • • • •
1
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sen(x)
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cos(x)
sen(x)sen(x) 6= 0
sec(x) :=1
cos(x)cos(x) 6= 0 csc(x) :=
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!"#" $%&'()&*+ ,-(.)&)/0,-('1+ 2* &3/*-)+ -*14*+ !!............................................................................................................................................................................................
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0 π• •
Cotangente"#$%&' ()*(+, -&./ '1 23 4'5$3543 6 74'5$3543 !"# % %&' ()*+* ! "#$ %&'( '* %*'+,('- +./.(0#$'( *,& 1'*,(.& #$+# '+,&!2'3 sen(390◦) !"# sen(480◦) ! # sen(570◦) !%# tan(390◦) !&# tan(480◦) !'# tan(570◦) !(# sen(−390◦) !)# sen(−480◦) !*# sen(−570◦) !+# sen(3π/2) !,# sen(5π/4) !-# sen(7π/6) !.# sen(−9π/4) !/# sen(−10π/3) !0# sen(−31π/12) !1# sen(5π) !2# sen(7π) !3# sen(9π) 4 51.2-&678 -19 &927 *19 &/ "-7/ 1 !7# :/ &- ;7861 ;7%87/6&< &- 9*(/1 %&- 19&/1 &9!"# :/ &- 6&8 &8 ;7%87/6&< &- 9*(/1 %&- 9&/1 &9! # :/ &- ;7861 ;7%87/6&< &- 9*(/1 %& -7 67/(&/6& &9!%# :/ &- 6&8 &8 ;7%87/6&< &- 9*(/1 %& -7 167/(&/6& &9!&# :/ &- ;7861 ;7%87/6&< &- 9*(/1 %& -7 19& 7/6& &9!'# :/ &- 6&8 &8 ;7%87/6&< &- 9*(/1 %& -7 9& 7/6& &9= >7--78 &- ?7-18 &@7 61 %& 19&/1< 9&/1< 67/(&/6& A 167/(&/6&< 2787 7%7 ;/1 %& -19 B/(;-19 x3;& 9& */%* 7/ 7 1/6*/;7 *C/< ;97/%1 -7 */'18.7 *C/ %7%7 !7# sen(x) = −1/2 A x &96B &/ &- 6&8 &8 ;7%87/6& !"# sen(x) = −1/2 A x &96B &/ &- ;7861 ;7%87/6&
! !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+.................................................................................................................................................................
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"#$%&' ()*(+, -&./ ' 12 34253 6 42 '572 " cos(x) = −5/13 # x $%&' $( %$)*(+, *-+.-(&$/ +" tan(x) = 3/4 # x $% *( '()*0, +$0 &$. $. *-+.-(&$/ $" cot(x) =√3 # x $% *( '()*0, +$0 1.23$. *-+.-(&$/ 4" cos(x) =
√2/2 # x $%&' $( $0 *-.&, *-+.-(&$/5/ 6$- α *( '()*0, -)*+, &-0 7*$ cos(α) = c/ 8(&,( $%9 -" cos(π − α) = :" sen(π − α) = " tan(π − α) = +" cos(π + α) = $" sen(π + α) = 4" tan(π + α) =;/ 6$- α *( '()*0, -)*+, &-0 7*$ cos(α) = c/ 8(&,( $%9 -" cos(π/2− α) = :" sen(π/2− α) = " tan(π/2− α) = +" cos(π/2 + α) = $" sen(π/2 + α) = 4" tan(π/2 + α) =</ 6$- α *( '()*0, -)*+, &-0 7*$ cos(α) = c/ 8(&,( $%9 -" cos(3π/2− α) = :" sen(3π/2− α) = " tan(3π/2− α) = +" cos(3π/2 + α) = $" sen(3π/2 + α) = 4" tan(3π/2 + α) = !"#" $%&'( ) *+,- '( &/ 01% 23%/( 42%3(32&'5/(8' 9:4;%21' 12 <312=34 <'72<.7# 34 >'&' =' >2# #1'1 534 ?' >&31% #13@ 257&2 37&'4@ '='4 A%5 #3524 B34253 6 C253* 8' $&./ '4 12 24'4 A%5 #3524 7#2525 =' A3&<' 12 351'4@ 13512%5' 351' 24 '=$3 ;%2 4%92 6 9'D'@ 12 <'52&' %5#A3&<2 6 &2>27#7#E'* F47'4 A%5 #3524 42
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. !"#$ %#$ &'()$*+('6 12 6 12
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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y = sen(2πt)N+&$1 8*+ *'#,$ t :' ,+1,+ 0 ' 26 +( B#3*($ 2πt '*&+#5' ,+ 0 ' >π6 ($ 8*+ +8*.:'(+ '*#' :*+(5' $ . ($ $&)(+5$; O )'%5.% ,+ +15' 0*# .D# /B1. ' )$,+&$1 $/5+#+% *#' 0*# .D#8*+ %+)%+1+#5+ *#' $#,' *'(8*.+% '#5.,', ,+ . ($1 )$% 1+3*#,$P /'15'%B &*(5.)(. '% +('%3*&+#5$ ,+ (' 0*# .D# '#5+%.$% )$% (' 0%+ *+# .' f ,+1+','6 )'%' $/5+#+%
y = sen(f · 2πt) = sen(2πft).Q$% +F+&)($6 $#,'1 $# 0%+ *+# .'1 >776 !77 = 77 . ($1 )$% 1+3*#,$6 1$# %+)%+1+#5','1 )$%y = sen(400πt)6 y = sen(600πt) = y = sen(800πt) %+1)+ 5.:'&+#5+;<' .#5+#1.,', ,+ (' $#,' +1 &'5+&B5. '&+#5+ (' '&)(.5*,; <' '&)(.5*, +1 (' ,+1:.' .D#&BG.&' 8*+ 5.+#+ (' $#,' %+1)+ 5$ ' *# :'($% )%$&+,.$; Q'%' *#' $#,' )+%0+ 5'6 +1 .3*'(' (' &.5', ,+ (' ,.0+%+# .' +#5%+ +( :'($% &BG.&$ = +( :'($% &9#.&$ '( '#@',$ )$% (' $#,';I$&$ ($1 :'($%+1 ,+ (' 0*# .D# /B1. ' y = sen(2πt) :'%9'# +#5%+ C2 = 26 +1' 0*# .D# 5.+#+ *#''&)(.5*, ,+ 2;E#' 0*# .D# 3+#+%'( $# '&)(.5*, |A| = $# 0%+ *+# .' f 1+%B +#5$# +1
y = A sen(2πft) +D.R$#,+ D %+)%+1+#5' '( :'($% +#5%'( $ )%$&+,.$P ($1 :'($%+1 ,+ y :'# ,+1,+ D − A6 4'15'D + A; <' S3*%' 2>;2T &*+15%' (' 3%BS ' ,+ *#' 0*# .D# $# '&)(.5*, 6 0%+ *+# .' ,+! . ($1 )$% 1+3*#,$1 = :'($% +#5%'( .3*'( ' +%$; U/1+%:+ 8*+ (' $#,' L)*#5+','M 8*+ ('' $&)'V' 5.+#+ &+#$% .#5+#1.,', )+%$ .3*'( 0%+ *+# .';
!"#" $%&'( ) *+,-./'( &0 -1%/.$%0( (.%$($.&'20( !"..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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t
y = 4 sen(6πt)
• • • • • • • •
1
4
−1 − 2
3− 1
3
2
31 4
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#$%&'( )*+),- ./01$2&3 4 5 #'6 &68 $( 4 !9&:$ (1/68265 ;(< =2'(: ('( 26'>:2$ (: 2(16: =/= 61 2$/?'6 36 &8 $8:2'&/682=5 (1 @&60=36/=: 36: '$?$' =/= 1( 0&'6A( 36 1( B'6 &68 $( 0'=3& $3(+ C:&(1/6826 :6 36:6(8 0'=3& $':$/&12D86(/6826 E('$(: 8=2(:+ F=' 6G6/01=5 &8 0$(8= 2$686 1( (0( $3(3 36 %686'('5 (1 =0'$/$'&8( H8$ ( 26 1(5 8= :I1= 1( B'6 &68 $( =''6:0=83$6826 ( 6:( 26 1(5 :$8= 2(/?$J8 (1%&8(:36 :&: ('/=8>(:5 6: 36 $' :=8$3=: &<( B'6 &68 $( :=8 &8 /H12$01= 6826'= 36 1( ='$%$8(1+9&:$ (1/6826 ;(?1(83=5 &8( = 2(E( /D: (12( 36 &8( 8=2( ?D:$ ( :6 2'(2( 36 &8( 8=2( @&62$686 &8( B'6 &68 $( $%&(1 (1 3=?16 36 1( ='$%$8(1+ C8( ='@&6:2( %686'( 2=3( &8( (82$3(336 B'6 &68 $(: < ('/=8>(: @&6 =/?$8(3=: 36 &8( B='/( 6:06 $(1 0'=3& 68 1= @&6 11(/(/=:/H:$ (K &(83= 1= (826'$=' 8= :6 1=%'( 2686/=: /D: = /68=: &8=: '&$3=: 6:0(82=:=:+L=: $8%68$6'=: 616 2'I8$ =: :=8 (0( 6: 36 0'=3& $' $' &$2=: @&6 %686'(8 &(1@&$6' B'6 &68 $(<5 68 0'$8 $0$=5 &(1@&$6' (/01$2&3 M3682'= 36 $6'2=: 1>/$26:N+ F=' 1= 2(82=5 0&6368 1=%'('O:$8262$A('P /H:$ (5 =/= = &''6 68 (0('(2=: 11(/(3=: O:$8262$A(3='6: 616 2'I8$ =:P+ Q(16: $' &$2=: :6 0&6368 (0'6 $(' 68 &(1@&$6' 26 1(3= 616 2'I8$ =5 3=836 (36/D: 36 1(: 8=2(:5 :60'=3& 68 1=: 2$/?'6: ('( 26'>:2$ =: 36 E('$=: $8:2'&/682=:+ !"#"!" $%&'( )*+ -./0'1%2-3 '(L(: =83(: 616 2'=/(%8J2$ (: 6:2D8 0'6:6826 36 &8( %'(8 (82$3(3 36 B='/(: (1'6363=' 8&6:R2'=+ L( 1&A E$:$?16 6: &8( 36 6:(: /(8$B6:2( $=86:5 &<=: =1='6:5 0=' 6G6/01=5 :6 36?68 (3$B6'6826: B'6 &68 $(:+ Q(/?$J8 2686/=: 1&A 8= E$:$?16 (1 =G= ;&/(8=5 =/= 1( 1&A $8B'(''=G(= &12'(E$=162(+ . 1= (826'$='5 36?6/=: (%'6%(' 1(: =83(: 616 2'=/(%8J2$ (: @&6 :$'E68 0('(1( =/&8$ ( $I8 36 :=8$3= M'(3$=N 6 $/D%686: M2616E$:$I8N+S6 36: &?'$I 6T06'$/682(1/6826 @&6 1(: =83(: 616 2'=/(%8J2$ (: 0=3>(8 (1 (8A(' %'(836:
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f(x) = 5 sen(350, 000πt)!798 g(x) = 3 sen(500, 000πt)!7 8 h(x) = −10 sen(200, 000πt)!:! ;)2) )') $-) '( &)* *+,$+(-.(* 1-')*< +-'+ )2 *$ /2( $(- +) 5 )43&+.$'! =$31-(2 >$( )')$-) '( (&&)* (* '( &) /124) y = A sen(2πft)!
!"#" $%&'()* $+,+%*- .+ /,* '/,)(0, 1(,212(.*- !" !" #" " %"
&' () *) +,-+. -,-/0+! %0 ..1%0)!%!-2 %,#*3!1 +!)*!4+0)/0 4.- 5!10- %0 6*) ,.)0- -,7*,0)/0-2,)%, !1 4! !+54,/*% 8 610 *0) ,! %0 !%! *)! %0 044!-' !" y = sen(x)2 y = sen(2x)2 0 ≤ x ≤ 2π' #" y = cos(x)2 y = cos(2x)2 0 ≤ x ≤ 2π' " y = sen(x)2 y = sen(x− (π/2))' %" y = sen(x)2 y = sen(x− π)' 0" y = sen(x)2 y = − sen(x)' 6" y = sen(x)2 y = sen(x) + 5' 7" y = cos(x)2 y = cos(x) + 1' 9" y = sen(x)2 y = 3 sen(x)2 ," y = sen(x)2 y = 3 sen(x) + 3' !"#" $%&' ) $*+*%), -* .+) /.+ 01+ 20+3430-),#$% &'( *+(,% y = A sen(Bx− C) +D - y = A cos(Bx− C) +D .,/.,%,(0$( 1$ &+.2$ 23%4,(,.$1 /+%*51, 6, '($ &'( *7( '-$ 4.38 $ ,% %*('%+*6$19:$6$ '($ 6, 1$% +(%0$(0,% + /$.32,0.+% ;', $/$., ,( ,( 1$ 6,8(* *7( 0*,(, '( %*4(*8 $6+/., *%+ ;', $($1*<$.,2+% $ +(0*('$ *7(9=1 /$.32,0.+ B (+% *(6* $ '31 ,% 1$ &., ',( *$ ;', ,%0$ 6$6$ /+. B/2π9 =%0+ (+% 6* ,*(2,6*$0$2,(0, ;', ,1 /,.>+6+ 0*,(, '( 1$.4+ *4'$1 $ 2π/B9 ?70,%, ;', ,1 !"#$ %& ! $'%! + /,.>+6+ ,% ,1 ., >/.+ + 6, 1$ &., ',( *$9
! !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+"# $%&'()*&+ A ,-., % #% %($#,*0. .) #% +-.%1 20) )3 ,40%# % |A|5"# $%&'()*&+ D .)3$#%6% #% +-.% 7)&*, %#()-*) 8 0-,.%.)3 (+7,)-.+ )# 7%#+& )-*&%#.) #% +-.% % y = D5 "3*) 7%#+& -+3 .% )# 7%#+& $&+().,+ .) 0- , #+ +($#)*+5 9+(+#% +-.% +3 ,#% %#&).).+& .) )3*) 7%#+&1 #+3 7%#+&)3 .) #% :0- ,;- )3*%&'- .)*)&(,-%.+3$+&D − |A| ≤ f(x) ≤ D + |A|."# $%&'()*&+ C ,-., % 0- .)3$#%6%(,)-*+ <+&,6+-*%# .) #% 4&'= %1 20) )- )# %3+ .)#%3 :0- ,+-)3 3,-03+,.%#)3 3) ##%(% )# !"#$"! .) #% :0- ,;-5%&!'()* +,-.-+- 8,>0?%& #% 4&'= % .) y = 4 sen(6πx)+25 @% :&) 0)- ,% )3 ,40%# % 6π/2π =
31 + 3)% 20) +()-6%&)(+3 .,>0?%-.+ y = sen(3x)5 A($#,%&)(+3 7)&*, %#()-*) )3% 4&'= %$+& 1 B #0)4+ 30>,&)(+3 *+.+ $+& C5...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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x
y=4 sen(3x)+2
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10.5
2
4
π2
π 2π
@% 4&'= % &)30#*%-*) *,)-) :&) 0)- ,% D1 $)&E+.+ 1/31 %($#,*0. B +3 ,#% %#&).).+& .) y = 25%&!'()* +,-.-,- F+& )?)($#+1 y = sen(x − π/4) )3 #% :0- ,;- 3)-+ !"#$"$ $ $+& π/40-,.%.)35 "- 0-% %-%#+4E% (03, %#1 3, y = sen(x) &)$&)3)-*% #% ()#+.E% 20) 3) )3*' %-*%-.+1y = sen(x− π/4) &)$&)3)-*%&' )# (,3(+ %-*+ 20) )($,)6% %*&%3%.+ $+& π/45%&!'()* +,-.-/- G-., %& #%3 $&,- ,$%#)3 %&% *)&E3*, %3 .) y = −5 sen(100πx) + 305H&) 0)- ,% I B
2π=
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2π= 501 $)&,+.+ I 1
f=
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505A($#,*0. I J5
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F = −kx VL(9 R( W&&X(Y'&$'( x (5 ,4 '+5%4$ +4 &A.+<+'4 & ($ &>+'4= k (5 #$4 &$5%4$%( *#( '(A($'( '(, <4%(.+4,'(, .(5&.%(2Z#A&$>4<&5 *#( 5( %+($( #$ .(5&.%( 5#Q(%& A&. #$ (H%.(<& 4 #$4 A4.(' 9 ($ (, &%.& (H%.(<&#$ #(.A& '( <454 m= &<& 5( +,#5%.4 ($ ,4 5+>#+($%( P>#.42), #(.A& 4A4.( ( [&%4$'& A4.4 +$'+ 4. *#( (, .& ( (5 '(5A.( +4-,(2Z#A&$(<&5 +$5%4,4'& #$ 5+5%(<4 N&.+O&$%4, '( &&.'($4'45 %4, *#( (, &.+>($ &+$ +'4 &$ ,4A&5+ +@$ ($ .(A&5& '( .(5&.%(2 Z( A.#(-4 ($ %(H%&5 '( ?I5+ 4= *#( #4$'& (5%+.4<&5 (, .(5&.%(A&. A 9 5&,%4<&5 (, #(.A&= T5%( &5 +,4.G 4,.('('&. '( ,4 A&5+ +@$ ($ .(A&5&= 9 ,4 ( #4 +@$'( 5# %.49( %&.+4 VA&5+ +@$ ($ ?#$ +@$ '(, %+(<A&Y %+($( ,4 ?&.<4
s = A cos(ωt), '&$'( ω =
√
k
m.
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3sen(t)5?5 y =
3
2cos(2t)5 @5 y = 4 cos(−t)5A5 y = 5 cos(4t). B5 y = −3 sen(4πt).C5 y = 6 sen(3πt). -5 y = −4 cos(π
2t). ! "#$ %&#'"()*$ +(" D *" ,B +(.(&)/!*& "* *)%"/.0+1 (" %(&/#+# = (" +($2*$( +( "* 20! /4! /!+/ *+*56&* ( 0!* 7&8: * +( "* 20! /4!5D5 y = sen(t− π
3)5 ,E5 y = 5 sen(3t+
π
6)5,,5 y = −2 sen(2πt) + 55 ,>5 y = −2 cos(2(t− π
8)) + 95,?5 y = −5 sen(2t− π
4)5 ,@5 y = 2 sen(3t+
π
4)5,A5 y = 2 cos(πt+ 3)5 ,B5 y = 3 sen(πt− 2)5,C5 F/ y = a sen(ωx − b) G*""*& (" %(&/#+# "* *)%"/.0+1 2&( 0(! /*1 %(&/#+# = +($2*$( (! .H&)/!#$+( a1 ω = b5,-5 I* (& 0!* *!8"/$/$1 $()(J*!.( *" +(" .(K.#1 +( "* 20! /4! f(x) = A cos(Bx− C) +D5,D5 L* %!+/+*+ +(" *70* (! 0! %0(&.# #)# 20! /4! +(" ./()%# ($.8 +*+* %#&
P = A sen(B(t− π
2)) + C+#!+( A ($ "* )/.*+ +( "* +/2(&(! /* (!.&( "* )*=#& = "* )(!#& %!+/+*+1 C ($ "* %!+/+*+%&#)(+/#1 = 2π/B ($ (" %(&/#+# +( "* )*&(*5 F0%#!/(!+# M0( (" %(&/#+# ($ +( +# ( G#&*$ =M0( "* *".0&* #! )*&(* *".* ($ +( >@ %/($ = #! )*&(* '*J* +( ,> %/($1 .&*N*& 0/+*+#$*)(!.(+#$ / "#$ +( "* 7&8: * +( "* 20! /4! P 5
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!"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+ ! "#$%&'(% &) *+%,( -& -& /,(0 (-( 1)( -& 0(' '/21/&)3&' ,&-/-('!4(5 35◦ 20′ 46′′! 465 55◦ 46′ 42′′! 4 5 68◦ 36′ 20′′!7! 8(00(% &0 +'&)+9 '&)+9 3()2&)3& : +3()2&)3& -& 0+' ;)210+' (21-+' -& 0+' 3%/;)210+' %& 3;)<210+' 1:+ (3&3+' '+) a : b9 : 1:( =/$+3&)1'( &' c!4(5 a = 59 b = 7! 465 a =√29 b = 1! 4 5 a =
√59 b = √3!4-5 a = 29 c = 5! 4&5 b =
√29 c = 1! 4*5 a = 19 c = 2!>! 8(00(% &0 +'&)+ : &0 '&)+ -& 1) ;)210+ (21-+ 1:( 3()2&)3& ,/-&4(5 1/2! 465 1! 4 5 2!4-5 0.3! 4&5 √
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2/3! 4&5 √3/3! 4*5 √
2/5!@! 8(00(% &0 '&)+ : 0( 3()2&)3& -& 1) ;)210+ (21-+ 1:+ +'&)+ ,/-&4(5 √2/2! 465 √
3/2! 4 5 1/3!4-5 0.1736! 4&5 0.4226! 4*5 0.9272!A! B'(% 0( /-&)3/-(- $/3(2C%/ ( $(%( $%+6(% D1&4(5 tan2(α) + 1 = sec2(x)! 465 1 + cot2(α) = csc2(α)!E! "#$%&'(% (-( %(FC) 3%/2+)+,G3%/ ( &) *1) /C) -& 0(' +3%('!HI! J%(F(% 0(' 2%;K (' -& 0(' '/21/&)3&' *1) /+)&'!4(5 y = 2 sen(x)! 465 y = 2| sen(x)|! 4 5 y = 2 sen(|x|)!4-5 y = 2 sen(x+ π/2)! 4&5 y = 2 cos(x)! 4*5 y = | cos(x)|!425 y = −2 sen(x)! 4=5 y = 2 sen(2x)! 4/5 y = −2 sen(4x)!4L5 y = cos(2x)! 4M5 y = −2 cos(2x)! 405 y = 2 cos(2x+ π)!HH! N/61L(% 0(' 2%;K (' -& 0(' '/21/&)3&' *1) /+)&'! 4O12&%&) /(P '1,(% 0(' yQ ++%-&)(-(' -& 0+''1,()-+'!54(5 y = cos(x) + sen(x)!465 y = cos(x) − sen(x)!4 5 y = x+ sen(x)!HR! N/61L(% 0(' 2%;K (' -& 0(' '/21/&)3&' *1) /+)&'! 4O12&%&) /(P ,103/$0/ (% 0(' yQ ++%-&)(-(' -&0+' '1,()-+'5(5 y = x sen(x)!65 y = x2 sen(x)!
!"#" $%$&'('()* +$, '-./01,) !" ! ! y = sen(5x) sen(x)"#$% &' ()'*+ P ,- .)-/- -' ,-'*01+ (+,0*0/+ ,+23- )'4 03 )'5-3-' 04 1- 3410+ 6 +' -'*3+ -'(0, 0) 7 +.(8-*4 )'4 /)-8*4 -' 2π/5 .0')*+,% 948843 5:3.)84, (434 84, ++31-'414, x 7 y 1-8()'*+" )4'1+ ,- ).(8- ;)-<4! P -,*= -' (5, 0) )4'1+ t = 0%<2! P -,*= -' (−5, 0) )4'1+ t = 0%< ! P -,*= -' (0, 5) )4'1+ t = 0%<1! P -,*= -' (−3, 4) )4'1+ t = 0%<-! P -,*= -' (0,−5) )4'1+ t = 0%#>% ?-4' P -8 (30.-3 ()'*+ (+,0*0/+ 1+'1- y = sen(x) *0-'- )' .=@0.+ 7 Q -8 (30.-3 ()'*+'-A4*0/+ 1+'1- y = sen(x)*0-'- )' .B'0.+% ?-4 O -8 +30A-' 1-8 ,0,*-.4 1- ++31-'414,%948843 84 (-'10-'*- 1- 84 8B'-4 ;)- (4,4 (+3 P 7 Q 7 1- 84 8B'-4 ;)- (4,4 (+3 P 7 O%#6% <C+'1) *4 -' 8+, -@*3-.+,!4! DE)F ()-1- 1- 03 1- 84 +'1) *4 -' ±∞ 1- 84 A3=G 4 1- y = sen(x)H2! DE)F ()-1- 1- 03 1- 84 +'1) *4 483-1-1+3 1- x = 0 1- 84 A3=G 4 1- y = sen(1/x)H &,43)'4 48 )841+34 A3=G 4 (434 0'*-'*43 /-3 84 A3=G 4 1- -,*4 5)' 0:'" (434 /48+3-, 1- x(3:@0.+, 4 I%#J% &,43 )'4 48 )841+34 A3=G 4 (434 (3+1) 03 84 A3=G 4 1- 84 3-84 0:' y = arctan(x)+arctan(1/x)%DE)F ()-1- +'K-*)343H L3+243 ,) +'K-*)34%#M% <N([email protected] 0+'-, (+80'+.048-,!% 9-.+, /0,*+ -' -8 *-@*+ ;)- 24,*4 +' +'+ -3 8+, /48+3-, 1-84, 5)' 0+'-, 03 )843-, -' -8 0'*-3/48+ [0, π/4] (434 +'+ -3 -8 /48+3 -' )48;)0-3 'O.-3+ 3-48 x%N +'*0')4 0:' 0'10 43-.+, 1+, P3- -*4,Q (+80':.0 4, (434 +.()*43 1- .4'-34 4([email protected]/48+3-, 1- sen(x) 7 cos(x)" )4'1+ |x| -, (-;)-R+% ! x ≈ 0 "#$%# "'
sen(x) ≈ x− x3
6+
x5
120
cos(x) ≈ 1− x2
2+
x4
24.S',4743 /48+3-, 1- x -' 0' 3-.-'*+, 1- 0.1 (43*0-'1+ +' 0.1 (434 1-*-3.0'43 T4,*4 ;)-/48+3-, 1- x 84, 5:3.)84, 1- 84 1-3- T4 (3-10 -' 8+, /48+3-, 1- +,-'+ 7 ,-'+ +' )' -33+3 '+.47+3 1- I%III# -' -8 /48+3%
! !"#$%&' ()* &!+ ,%- .'-/+ $0.1'-'23$0. !+
!"#$%&' () !"#$%$&' !() )%)*( "+) !" " #$% '(%)* +,'-(%(./0,' 1* 2%3),*1* !"! #$% &'( *!(+% ,-*.!(!"/0-* $% %!( 1+-*23* $% 45 1'+3+( 0+(+- *(6+-%$%7 8$ 9'+ :$8*(;(*0!% 6$#!-+% 3+# 3!"*(*! 9'+ 1-!3' +( +# "*%"! 6$#!- 1$-$ #$ &'( *2(< =*( +">$-.!7%* 9'+-+"!% %!#' *!($- + '$ *!(+% 3!(3+ #$ *( 2.(*0$ $1$-+? $ +( +# $-.'"+(0! 3+ '($&'( *2( 0-*.!(!"/0-* $ (+ +%*0$"!% @*(6+-%$%A< B$ "$(+-$ 3+ #!.-$- #! $(0+-*!- +% "+3*$(0+'( !"1-!"*%!< C-*"+-! %+#+ *!($"!% '( *(0+-6$#! 3!(3+ #$ &'( *2( -+%0-*(.*3$ 1'+3$ 0+(+-*(6+-%$7 8 $ #$ *(6+-%$ 3+ +%$ -+%0-* *2( %+ #+ ##$"$ !( *.'$# (!">-+ 9'+ #$ &'( *2( 1+-!$(0+1!(*+(3! ! 3+#$(0+ 3+# (!">-+<#$ %! &'()&D# $- !%+(! +% #$ &'( *2( *(6+-%$ 3+ #$ -+%0-* *2( 3+ #$ &'( *2( %+(! $# *(0+-6$#! [−π/2, π/2]8 0+(+"!% $%E '($ &'( *2(arc sen : [−1, 1] −→ [−π/2, π/2]0$# 9'+ y = arc sen(x) 9'*+-+ 3+ *- 9'+ arc sen(x) +% +# F(* ! G(.'#! +( +# *(0+-6$#! [−π/2, π/2] '8! %+(! +% *.'$# $ x< D% 3+ *-7
y = arc sen(x) ⇐⇒ sen(y) = x, −π/2 ≤ y ≤ π/2.B$ ;.'-$ HI<H "'+%0-$ #$ .-G; $ 3+ #$ &'( *2( $- !%+(!< B++-+"!% arc sen(x) !"! @+# G(.'#!J$- !K '8! %+(! +% xA<L#.'(!% $'0!-+%7 8 +%1+ *$#"+(0+ #$% $# '#$3!-$%7 '%$( #$ (!0$ *2( @sen−1(x)A +( #'.$- 3+$- !%+(! 1$-$ *3+(0*; $- #$ &'( *2( 9'+ $ $>$"!% 3+ 3+;(*-< B$ (!0$ *2( 1'+3+ 1-+%0$-%+$ !(&'%*!(+% 8$ 9'+ $K %+(! (! 0*+(+ *(6+-%$ >K 1!3-E$ !(&'(3*-%+ !( 1/ sen(x)7 8$ 9'+0$">*/( '%$"!% +%$ (!0$ *2( 1$-$ *(3* $- 1!0+( *$%<
! !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, !• ••
−π/2 π/2
1
y=sen(x)
................................................................................................................................................................................................................
−1 1• ••
π/2
−π/2
y=arcsen(x)
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"#$%&' ()*(+ ,' -%. #0. 1& 234.2 ! "# % %&'(%,' -%. #0. )# % %&'(% 34 546.4 54 7'.4&' 34748'.94 ': '& 234.2; 2. :' 5#-4&4. #' 54 <%434:4 #2.'723 272 #.94&=':2 >?3# 2 ' [0, π] 4. :%$'& 54 [−π/2, π/2]*y = arc cos(x) ⇐⇒ cos(y) = x, 0 ≤ y ≤ π.,44723 arc cos(x) 272 4: ?.$%:2 %@2 234.2 43 x*
• •π
1
y=cos(x)
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−1 1• ••
π
y=arccos(x)
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"#$%&' ()*A+ ,' -%. #0. 1& 2 234.2*) "# %+)(,'(+'"#.':74.94 2.3#54&'723 :' '& 29'.$4.94 <%4 43 546.#5' B2&y = arctan(x) ⇐⇒ tan(y) = x, −π/2 < y < π/2.,' $&?6 ' 34 7%439&' 4. :' 6$%&' ()*)* ,44723 arctan(x) 272 C4: ?.$%:2 %@' 9'.$4.9443 xD* E3%':74.94 .2 34 546.4. :'3 '& 2 29'.$4.94; '& 234 '.94 @ '& 2 234 '.94 @' <%4 2.'& 2 234.2; '& 234.2 @ '& 29'.$4.94 B254723 9&'>'8'& 92523 :23 B&2>:47'3*
!" " #$%&'(%)* +,'-(%(./+,'&0* '%1),*0* !• ••−π/2 π/2
1
y=tan(x)
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−10 10• ••
π/2
−π/2
y=arctan(x)
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"#$%&' ()*)+ ,' -%. #0. 1& 23'.$4.34 !"# % %&' ()*(* ! "#$%&$' ()* &($' $% &%$,-'$!.$/ arc sen(0)! .0/ arc sen(1)!. / arc cos(0)! .,/ arc cos(1)!.1/ arctan(0)! .2/ arctan(1)!.3/ arc sen(√2/2)! .4/ arc cos(1/2)!.)/ arc sen(−1/2)! .5/ arc cos(−1/2)!.6/ arctan(−1)! .%/ arctan(−
√3)!7! 8'-0$' 9&1.$/ arc sen(−x) = − arc sen(x)! .0/ arctan(−x) = − arctan(x)!:! ;)<=%)> $' ? 1@='1($' 1* AB'<)*-( ,1 x!.$/ sen(arc cos(x))! .0/ tan(arc cos(x))!. / cos(arc sen(x))! .,/ tan(arc sen(x))!.1/ sen(arctan(x))! .2/ cos(arctan(x))!.3/ sen(2 arc sen(x))! .4/ cos(2 arc sen(x))!C! 8'-0$' 9&1 arc cos(x) + arc sen(x) = π/2!D! E'$F$' %$( 3'G> $( ,1 %$( ()3&)1*A1( 2&* )-*1(!.$/ y = tan(x− π/2)! .0/ y = tan(−x)!. / y = tan(2x)! .,/ y = arctan(−x)!H! .I-*,& A$ 1* %-( 1@A'1<-(J KL&B =&1,1 ,1 )'(1 ,1 %$ 3'G> $ ,1 y = arctan(x) &$*,- x →
±∞J
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! !"#" $%&'()*+ ,&-./0/'12&- *+#$ %&'(&)*+) ,-.,.%& &0* *.% 1$-2 (*$ 30-4(5$ ,$-$ )5 .%)*. 6 )5 %)*. 1) 5$ %(4$ 1)1.% 2*'(5.%7 8$-)4.% (* $-'(4)*+. ').49+-& . ,$-$ .*:)* )-*.% 1) ;()< )* )5 $%. 1)2*'(5.% $'(1.%< .=+)*)4.% 5$% -$>.*)% +-&'.*.49+-& $% ,$-$ 5$ %(4$ 1) 5.% 2*'(5.%7 #()'.<%(,.*)4.% ;() +$5 -)5$ &0* )% :25&1$ ,$-$ ($5;(&)- :$5.- 1) 5.% $-'(4)*+.%7?* 5$ @'(-$ AB7 )5 %)'4)*+. OF 3.-4$ (* 2*'(5. α .* )5 )C) 1) X< 4&)*+-$% ;() )5%)'4)*+. OG .* )5 %)'4)*+. OF 3.-4$ (* 2*'(5. β7 D-.=$-)4.% ;() 5$% ..-1)*$1$% 1)5,(*+. G )%+2* 1$1$% ,.-xG = cos(α) cos(β)− sen(α) sen(β) 6 yG = sen(α) cos(β) + cos(α) sen(β). !"#$%&' (&"& %& )$#&* E.4. )5 2*'(5. ;() 3.-4$ OG .* )5 )C) X )% &'($5 $ α+ β<5$% -)5$ &.*)% $*+)-&.-)% *.% 1$* 5.% :$5.-)% 1)5 .%)*. 6 )5 %)*. ,$-$ 5$ %(4$ 1) 5.% 2*'(5.%7
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..........................................................................................................................................................................
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•• •
•• •
•
•
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............. ............. .............
A B
C
DE
F
G
0
AG⊥OB
BE⊥OB
EG⊥OF
F&'(-$ AB7 G H$>.*)% +-&'.*.49+-& $% ,$-$ 5$ I(4$?5 -)%(5+$1. 1),)*1) 1) 5$ .=%)-:$ &0* 1) ;() )5 2*'(5. ∠CED )% &'($5 $5 2*'(5. α7 E$5J (5)4.% xG 6 yG7xG = OA = OB −AB = OB −DE
= OE cos(α) − EG sen(α)
= OG cos(β) cos(α) −OG sen(β) sen(α)
= cos(α) cos(β)− sen(α) sen(β)
yG = AD +DG = EB +DG
= OE sen(α) +EG cos(α)
= OG cos(β) sen(α) +OG sen(β) cos(α)
= sen(α) cos(β) + cos(α) sen(β)?*+.* )%< ,.1)4.% 1) &- ;()cos(α+ β) = cos(α) cos(β)− sen(α) sen(β)
sen(α+ β) = sen(α) cos(β) + cos(α) sen(β).D.- 5. ;() $@-4$4.% ;()
!"#" $%&'()*+ ,&-./0/'1,&-2*+ !" !"#"$% %'( )*+,+) #$%&'()*+ ,- .,/ /%12+ !"# s $ t #%&!'() '!"*!) ,"*!)-,.!'"/ 0#12(# !)3cos(s+ t) = cos(s) cos(t)− sen(s) sen(t)
sen(s+ t) = sen(s) cos(t) + cos(s) sen(t).3*+ 4%&'()*+ *15-&/6&-+ +61 )*+ 4%&'()*+ 78+/ *+ 9*&* 5&*7*:*& 61 )*+ 4(1 /61-+ 6+-16 ; +-16<=6,*+ )*+ ,-'8+ &-)* /61-+ ,-9-1,-1 ,- -))*+< >?*'/1*&-'6+ *)@(1*+ ,- )*+ 61+- (-1 /*+<-'!./01$ #1!1 01 2%34!4( %1+cos(s− t) = cos(s+ (−t))
= cos(s) cos(−t)− sen(s) sen(−t)= cos(s) cos(t) + sen(s) sen(t).A61,- -1 )* B)5/'* )C1-*D (+*'6+ )*+ &-)* /61-+ ,- 9*&/,*, ,- )*+ 4(1 /61-+< .18)6@*'-15-D
sen(s − t) = sen(s+ (−t))= sen(s) cos(−t) + cos(s) sen(−t)= sen(s) cos(t)− cos(s) sen(t).E6,-'6+ &-+('/& )*+ 4%&'()*+ ,- *,/ /%1 675-1/,*+ F*+5* *G(C 96&
cos(s± t) = cos(s) cos(t)∓ sen(s) sen(t)
sen(s± t) = sen(s) cos(t)± cos(s) sen(t)3*+ 4%&'()*+ *15-&/6&-+D 61:(15*'-15- 61 )6+ H*)6&-+ ,- 6+-16 ; +-16 9*&* H*)6&-+ -+9- /*I)-+ 6'6 π/2, π, 3π/2, 2πD 16+ 9-&'/5-1 &- (9-&*& )*+ 4%&'()*+ G(- (+*'6+ -1 )*+ +- /61-+*15-&/6&-+ 9*&* 6'9(5*& )6+ H*)6&-+ ,- )*+ 4(1 /61-+ -1 )6+ ,/4-&-15-+ (*,&*15-+ * 9*&5/& ,-H*)6&-+ -1 -) 9&/'-& (*,&*15-< J6+ 9-&'/5-1 *,-'8+ 6'9(5*& ,- 46&'* -?* 5* )6+ H*)6&-+,- )*+ 4(1 /61-+ 9*&* 65&6+ 81@()6+<564.#0" )*+,+)+cos(π/2 + x) = cos(π/2) cos(x)− sen(π/2) sen(x) = 0 · cos(x)− 1 · sen(x) = − sen(x).
sen(π/2 + x) = sen(π/2) cos(x) + cos(π/2) sen(x) = cos(x).3* B)5/'* &-)* /%1 675-1/,* '(-+5&* G(- )* @&8K * ,- 6+-16 -+ +/16+6/,*)D ;* G(- -+ )*@&8K * ,- +-16 ,-+9)*L*,* F* /* )* /LG(/-&,* 96& π/2 (1/,*,-+<564.#0" )*+,+,+ M*))*& -) H*)6& -?* 56 ,-) 6+-16 ,- (1 81@()6 ,- "!◦#N π/122<5-1-'6+ G(- 15 = 45 − 30<cos(15◦) = cos(45◦ − 30◦) = cos(45◦) cos(30◦) + sen(45◦) sen(30◦)
=
√2
2·√3
2+
√2
2· 12=
√2
4(√3 + 1).
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! !"#$%&' () *+,%) -./$%+0 #$%&'( )%$ *+,-.)%$ %&/0,1(,0$ /0&0-($ 2.0cos(2x) = cos(x+ x) = cos(x) cos(x)− sen(x) sen(x) = cos2(x)− sen2(x).
sen(2x) = sen(x) cos(x) + cos(x) sen(x) = 2 sen(x) cos(x).3%$ ,0)% 1(&0$ '0 5&6.)( '(7)0 1).$/,%& 8%,1%$ ($%$ 1&/0,0$%&/0$9 3% ,0)% 1+& 12 sen(2x) =
sen(x) cos(x) &($ '1 0 2.0 0) :,('. /( '0 )%$ *.& 1(&0$ $0&( ; ($0&( 0$ &.08%-0&/0 .&%*.& 1+& $1&($(1'%) 2.0 /10&0 0) '(7)0 '0 *,0 .0& 1% 2.0 )%$ *.& 1(&0$ (,161&%)0$< :0,( (& )%-1/%' '0 )% %-:)1/.'9 =>?%)( 0& .&% %) .)%'(,% 6,5@ %A9B(, $. :%,/0< .$%&'( )% ,0)% 1+& B1/%6+,1 %< :('0-($ ,00$ ,171, )% ,0)% 1+& cos(2x) = cos2(x)−sen2(x) '0 .&% '0 )%$ '($ -%&0,%$ $16.10&/0$
cos(2x) = cos2(x)− (1− cos2(x)) = 2 cos2(x)− 1 =CD9CAcos(2x) = (1− sen2(x)) − sen2(x)) = 1− 2 sen2(x) =CD9"AE&/(& 0$ :('0-($ ,0$.-1, )%$ *+,-.)%$ '0 '(7)0 5&6.)( (-(
cos(2x) = cos2(x)− sen2(x)
cos(2x) = 1− 2 sen2(x)
cos(2x) = 2 cos2(x)− 1
sen(2x) = 2 sen(x) cos(x)F0$:0G%&'( cos2(x) ; sen2(x) 0& )%$ 0 .% 1(&0$ %&/0,1(,0$ (7/0&0-($ 2.0cos2(x) =
1 + cos(2x)
2, sen2(x) =
1− cos(2x)
2=CD9DAH00$ ,17%-($ )% 0 .% 1+& CD9D
cos2(x) =1 + cos(2x)
2=
1
2+
1
2cos(2x).IJ+-( $0,5 )% 6,5@ % '0 y = cos2(x) !" #$%" '() ")*$#'+# )+, -' $ (.+ ,$ /0$-$ +1*$)$#%" 2#34 " -$ y = cos2(x)5 6"#*'.+, -$ %" 2#34 " -$ y = cos(2x)7 .0%*'/%' ")-+ ,0 "./%'*0-/+# 89:;< /"#" +1*$)$# y =
1
2cos(2x) = -$,/%">")-+ ?$#*' "%.$)*$ %" 2#34 " ")*$#'+# /+#9:; @" '" "##'1"5 A% #$,0%*"-+ ,$ '%0,*#" $) %" ,'20'$)*$ 420#"5
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............
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x
y=cos2(x)
......................................................................................................................................................
.......................................................................................................................................................................
.................................................................................................................................
..................................................................................................................................................1
0.5
π2
π 2π
!"#" $%&'()*+ ,&-./0/'1,&-2*+ !"#$ %& ()* y = cos2(x) &$ +,-.(/0 10, 210 (30 $(04$4(%,5* 40 2)& 1&0 (, %4.5& , 5, 210 (30.6$( , 7 %& ,-85(+1% 9:;<= ,5)&%&%4) %& 5, 5>0&, y = 1/2? !"#$%&' () *)(+, -./$%,0 @& 5,$ 23)-15, 9:"?"= 4.+&0&-4$* $1$+(+17&0%4 x/2 84) x*A1& B3)-15,$ %&5 C0D154 E&%(4cos(
x
2) = ±
√
1 + cos(x)
2sen(
x
2) = ±
√
1− cos(x)
2.12)#3%, 4506050 F,55,) 54$ G,54)&$ &H, +4$ %& cos(π/8) 7 sen(π/8)? !"#$% '()* I4-4$,.&-4$ A1& cos(π/4) =
√2/2* +&0&-4$ A1&cos(
π
8) =
√
1 +√2/2
2=
√
2 +√2
2.J0654D,-&0+&* +&0&-4$ A1&
sen(π
8) =
√
1−√2/2
2=
√
2−√2
2. !"#" " $%&'()* (%),- .(),*.,- /, 01) (%),- %) (21*. 03, 1,) (*7",8%)#& 45040 +#)",-%'- $. /-01 . 2! y = 3 sen(x) + 4 cos(x)* !"#$% '()* K1840D,-4$ A1& y 21&), %& 5, 24)-, y = C sen(x+ θ)? L&0%)>,-4$ A1&
C sen(x+ θ) = Csen(x) cos(θ) + C cos(x) sen(θ) = 3 sen(x) + 4 cos(x).I4-8,),0%4 54$ 4&M (&0+&$ %& 4$&04 7 $&04 &0 5, (D1,5%,% ,0+&)(4)* G&-4$ A1& +&0&-4$A1&C cos(θ) = 3 9:= C sen(θ) = 4 9<=.I1,%),0%4 &0 9:= 7 9<= 7 $1-,0%4 4.+&0&-4$ A1&C2 cos2(θ) + C2 sen2(θ) = 32 + 42 = 52.N $&, A1& C = ±5? @(G(%(&0%4 -(&-.)4 , -(&-.)4 9<= 84) 9:=* 4.+&0&-4$ A1&
C sen(θ)
C cos(θ)= tan(θ) = 4/3
! !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, !"#$%&' (&)$*&+ ,-.*,. /#$y = 5 sen(x+ θ), &1 θ = arctan(4/3).2&. 3& /#$ 3, %.4- , )$ y $+ #1, 5#1 671 +61&+&6),3 )$ ,*(368#) !' 6%#,3 5.$ #$1 6, /#$ 3,+5#1 6&1$+ &.6%61,3$+ ($.& )$+5,+,), (&. −θ9 ", %.4- , +$ 63#+8., , &1861#, 6719
:6%#., ;<9!= ", %.4- , )$ y = 3 sen(x) + 4 cos(x)9>3 .,?&1,*6$18& #+,)& $+ %$1$.,39 2&. 3& /#$ 8$1$*&+ 3, +6%#6$18$ (.&(&+6 671' #@, )$*&+A8., 671 )$B,.$*&+ ,3 3$ 8&.9 !"#"$% %'( )*+,+, CD&*E61, 671 361$,3 )$ F$1&+ @ D&+$1&+G+ ! "#$ &'$ y = A sen(x) +B cos(x)( )#*+* *, -&.&-,* /0/
y = C sen(x+ θ) /$ C =√
A2 +B2 1 θ = arctan(B/A)2 !"#"#" $%&' )*+%, -% ./+ )*+%, *+ 0),1)+1', .2% /%+ )',- .%/%010 )*+)+ H+,. #1, ,3 #3,)&., %.4- , (,., .$(.$+$18,. 3,+ +6%#6$18$+ 5#1 6&1$+;9 I6E#B,. $1 #1, *6+*, (,18,33,' y = 0.5x' y = −0.5x' y = 0.5x · sen(x)9 >+ .6E, #1,)$+ .6( 671 )$ 3& /#$ J$' )$+861,), (,., #1, ($.+&1, /#$ 1& K,@, J6+8& 3, %.4- ,9L9 I6E#B,. $1 #1, *6+*, (,18,33,' y = 0.1x2' y = −0.1x2' y = 0.1x2 · sen(2x)9 >+ .6E,#1, )$+ .6( 671 )$ 3& /#$ J$' +#(&16$1)& /#$ $+84 )$+861,), (,., #1, ($.+&1, /#$ 1&K,@, J6+8& 3, %.4- ,9<9 y = 5 sen(0.4x) · sen(10x) ", %.4- , .$+#38,18$ &..$+(&1)$ , #1, ,*(368#) *&)#3,),9 9 y = 5 sen(0.4x) + sen(10x)9!9 y = 4cos(3x) + 3 sen(2x)9
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cos(x+ y) = cos(x) cos(y)− sen(x) sen(y) DEF4 Gcos(x− y) = cos(x) cos(y) + sen(x) sen(y) DEF4!GH0/,1*# %$,$ % 0, )#1%$6 #=-%1%/#$ 70%cos(x+ y) + cos(x− y) = 2 cos(x) cos(y)/)%1-',$ 70% '%$-,1*# +, 5')/%', *% +, $%801*,6 #=-%1%/#$ 70%cos(x− y)− cos(x+ y) = 2 sen(x)sen(y).",$ '%+, )#1%$ ,1-%')#'%$ 1#$ /0%$-',1 #/# #1.%'-)' 5'#*0 -#$ %1 $0/,$ 9 '%$-,$4 !"#$%& '()*)+) cos(3x) cos(5x) = 1
2[cos(8x) + cos(2x)]4I#+# ,1*# u = x+ y 9 v = x− y6 $% -)%1% 70% x = (u + v)/26 /)%1-',$ 70% y = (u − v)/24@#' +# -,1-#6 +,$ '%+, )#1%$ ,1-%')#'%$ 50%*%1 '%%$ ')=)'$% #/#
cos(u) + cos(v) = 2 cos(u+ v
2) cos(
u− v
2).
cos(v)− cos(u) = 2 sen(u+ v
2) sen(
u− v
2). !"#$%& '()*),) cos(5x) + cos(3x) = 2 cos(4x) cos(x)4 !"#$%& '()*)-) C%$#+.%' +, % 0, )31 sen(7x) = sen(5x)4 !"#$% '()* C% #'*,1*# 70% α 9 π−α -)%1% )80,+ .,+#' *% $%1#6 ,$J #/# $0/, *% /<+-)5+#$*% 2π *% %$#$ >180+#$6 -%1%/#$ 70% +, % 0, )31 *,*, )/5+) , 70%
7x = 5x+ 2kπ, # DEG7x = π − 5x+ 2nπ = −5x+ 2kπ. DKG
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, !#$ %&' ()*$+$,(- ./$ 2x = 2kπ ( -$0 x = kπ1 (+ k 03)4*3034( 5$ 5(+5$ 2x = 2kπ6 *$+$,(-./$ x = kπ6#$ %7' ()*$+$,(- ./$ 12x = (2k + 1)π6 8/$9(1 x = (2k + 1)π/126 : -$0 ./$ *(50- ;0--(;/ 4(+$- -(+ x = kπ ( x = (2k + 1)π/121 k 03)4*3034(6 !"# % %&' ()*+* ! "#$%&' ($# )*+,-($# &. #-,$ ' +.#/$ &. $+0-,.%/'#1 2$(($+ cos(x)1 sen(x) &'%&. x 3-.&. #.+π/2± α1 ' π ± α1 3π/2± α! "#$+ .#$# )*+,-($# 3$+$ ',3(./$+ ($ #50-5.%/. /$6($!
α π/2− α π/2 + α π − α π + α 3π/2− α 3π/2 + αcos − sen(α)sen cos(α)tancotseccsc7! 8+'6$+ $&$ -%$ &. ($# #50-5.%/.# +.($ 5'%.#!9$: cos(3α) = cos3(α)− 3 cos(α) sen2(α) = 4 cos3(α)− 3 cos(α)!96: sen(3α) = 3 cos2(α) sen(α) − sen3(α) = 3 sen(α) − 4 sen3(α)!9 : tan(α+ β) =
tan(α) + tan(β)
1− tan(α) tan(β)!9&: tan(α− β) =
tan(α)− tan(β)
1 + tan(α) tan(β)!9.: tan(2α) =
2 tan(α)
1− tan2(α)!;! <% -% ,5#,' #5#/.,$ &. ''+&.%$&$#1 &56-=$+ ,$%-$(,.%/. ('# '%=-%/'# &. )-% 5'%.# #5>0-5.%/.#1 5%&5 $+ ($ $,3(5/-& ? )+. -.% 5$ &. $&$ -%$ &. .(($#!9$: y = sen(x)1 y = sen2(x)! 96: y = cos(x)1 y = − cos(2x)!9 : y = cos(x)1 y = −(1/2) cos(2x)! 9&: y = sen(x)1 y = 2 sen(3x) + 5!@! <% ('# #50-5.%/.# .=.+ 5 5'#1 -#$+ ($ 5%)'+,$ 5*% &$&$ 3$+$ 2$(($+ 95: cos(t/2)1 955: sen(t/2) ?9555: tan(t/2)!9$: sen(t) = 12/131 π/2 < t < π! 96: cos(t) = 4/51 3π/2 < t < 2π!9 : tan(t) = 21 π < t < 3π/2!A! <% ('# #50-5.%/.# .=.+ 5 5'#1 -#. ($ 5%)'+,$ 5*% &$&$ 3$+$ 2$(($+ 95: cos(2t)1 955: sen(2t) ? 9555:
tan(2t)!9$: sen(t) =√2/31 π/2 < t < π!
!"#" $%&'()*+ ,&-./0/'1,&-2*+ !" !" cos(t) =√3/5# 3π/2 < t < 2π$ " tan(t) = 1/2# π < t < 3π/2$&$ '())(* +) ,()-* +.( /- 0+ (0( 12( 0+ )(3 +.4*+35-2+3 35615+2/+3$ (" cos(π/12) !" sen(π/12) " sen(3π/8) 0" cos(3π/8) +" tan(15◦) 7" cot(112.5◦)8$ 9*-!(* 13(20- )-3 :;/-0-3 0+ )( 3+ 5<2 =>$?$=" @1+ (0( 12( 0+ )(3 35615+2/+3 712 5-2+3 +335213-50()$ '())(* 31 (:4)5/10 A 0+37(3+$ B*(C(* )( 6*DE ($ (" y = sen(x) + cos(x)$ !" y = sen(x)− cos(x)$ " y = 2 sen(2x) + cos(2x)$ 0" y = 3 cos(x)− 2 sen(x)$F$ G.4*+3(* (0( 31:( +2 )( 7-*:( C cos(x− θ) (" √
3 cos(x) + sen(x)$ !" 2 cos(x) − 2 sen(x)$ " 4 sen(5x)− 3 sen(x)$ 0" √2 cos(x) −
√2 sen(x)$H$ 9*-!(* )(3 35615+2/+3 50+2/50(0+3 (" (cos(t) + sen(t))2 = 1 + sen(2t)$ !" cot(2θ) =
1
2(cot(θ)− tan(θ))$ " sec(2x) =
1
2 cos2(x) − 1$ 0" cos(2u) =
cot(u)− tan(u)
cot(u) + tan(u)$ +" tan(
x
2) =
sen(x)
1 + cos(x) 7" cot(2x) =
1 + cos(2x)
1− cos(2x)$ 6" sen(4x) = sen(2x)(cos2(x)− sen2(x))$ I" sen(4x) = 4 sen(x) cos(x)− 8 sen3(x) cos(x)$ 5" cos(4x) = 8 cos4(x)− 8 cos2(x0 + 1$ J" sen(x+ y) + sen(x− y) = 2 sen(x) cos(y)$ K" cos(x+ y) + cos(x− y) = 2 cos(x) cos(y)$ )" cos4(x) =
1
8(3+ 4 cos(2x) + cos(4x)) L16+*+2 5(M 13(* )(3 7<*:1)(3 0+) D261)- :+05-# 0-3,+ +3"$ :" sen4(x) =
1
8(3− 4 cos(2x) + cos(4x))$=N$ O3(* )(3 7-*:1)(3 0+ (05 5<2 A *+3/( 4(*( 3+2-# 4(*( 4*-!(* @1+
2 sen(x) cos(y) = sen(x+ y) + sen(x− y)A @1+sen(u) + sen(v) = 2 sen(
u+ v
2) cos(
u − v
2), sen(u)− sen(v) = 2 cos(
u+ v
2) sen(
u− v
2).
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! ! "##$ &'('& )*$ $'+,'#-.#$ /&*0, .*$ *1* $,12$!324 sen(9x) sen(5x)! 3(4 cos(9x) cos(7x) cos(5x)! 3 4 sen(9x) sen(7x) sen(5x) sen(3x)! 5! "##$ &'('& )2$ $'+,'#-.#$ $,12$ *1* /&*0, .*$!324 y = sen(x) cos(2x)!3(4 y = 4 sen(x
2) sen(
7x
2)!3 4 y = 8 sen(
5x
2) cos(
3x
2)! 6! 7&*(2& 202 ,-2 0# )2$ $'+,'#-.#$ '0#-.'020#$!324 cos(2x) = −2 sen(x+π
4) sen(x− π
4)!3(4 sen(2x) = 2 sen(x+
π
2) cos(x− π
2)!3 4 tan(4x) =
sen(x) + sen(7x)
cos(x) + cos(7x)!304 tan(2x) =
sen(x) + sen(2x) + sen(3x)
cos(x) + cos(2x) + cos(3x)!3#4 tan(x+ y)
tan(x− y)=
sen(2x) + sen(2y)
sen(2x)− sen(2y)!384 sen(3x)− sen(x)
cos 3x− cos(x)= −cot(2x)!3+4 sen(5x)− sen(x)
cos(5x) + cos(x)= tan(2x)!394 sen(x) + sen(y)
cos(x) + cos(y)= tan
(
x+ y
2
)!3'4 sen(x) + sen(y)
cos(x)− cos(y)== cot
(
x− y
2
)!3:4 cos(x)− cos(y)
sen(x)− sen(y)= − tan
(
x− y
2
)! ;! cos(x) + cos(2x) + cos(3x) = (1 + 2 cos(x)) cos(2x)! <! sen(x) + sen(2x) + sen(3x) = (1 + cos(2x)) sen(2x)! !"!" # %& '()*+ ,-'.()(/01-' &+#$ %&'( &% *+$ %&',-*(.%/0& 1( .%&01, *+$ -% 1(& 11(/(-(& "# $%& ' ()$*%&%+,()$ #'2 #$'(1%& % ,( *0$%&3 1( *$ +4$*'( %& ,$ 5$4,10 64%$%.(1*7(-08 9 10& -('0& %&'5$ %$ ':./*$0& -%1(& ;,$ *0$%& *. ,1(.%&2
!"!" #$%&$'()#* +,'-()(./+,'$&* !" !"#$%& '()()') #$%%$& '()(* %(* +,-.&(* x '$%.* /0. sen(x) = 1/21 !"#$% '()* 2$3.-(* ).*). %(* 4+4 4(* ). +0.*'&(* .*'0)4(* '&46(+(-7'&4 (* /0. .% 8+60%(). 9:◦ '4.+. 0+ *.+( 460$% $ 1/21 ;(-( 9:◦ .+ &$)4$+.* .* 460$% $ π/6< '.+.-(* /0. 0+$*(%0 4=+ ). %$ . 0$ 4=+ $+'.&4(& .* x = π/61>?* .*$ %$ ,+4 $ *(%0 4=+ ( @$3&8 ('&$* *(%0 4(+.*AB$ &.*C0.*'$ .* /0. '.+.-(* -0 @$* *(%0 4(+.* C(*43%.*< ). @. @( 4+D+4'$*< E$ /0. (-( %$F0+ 4=+ '4.+. C.&G()( 2π< '.+.-(* /0. $)$ 2π *. &.C4'. *0 H$%(&< %( /0. +(* )4 . /0.sen(
π
6+ k · 2π) = 1
2,C$&$ 0$%/04.& H$%(& .+'.&( ). k1 I(& %( '$+'(< %$ . 0$ 4=+ )$)$ '4.+. 4+D+4'$* *(%0 4(+.*
x =π
6+ 2kπ, k = 0,±1,±2, . . .>J.+)&8 ('&$* *(%0 4(+.*AB$ &.*C0.*'$ .* +0.H$-.+'. $D&-$'4H$< E$ /0. sen(π − α) = sen(α)< %( /0. +(* )4 . /0.
sen(5π
6) = sen(π − π
6) = sen(
π
6) =
1
2.I(& %$ C.&4()4 4)$) ). %$ F0+ 4=+ *.+(< '.+.-(* ('&$ F$-4%4$ 4+D+4'$ ). C(*43%.* *(%0 4(+.*12. H.&4D $ /0. π/6 E 5π/6 *(+ %$* ,+4 $* *(%0 4(+.* ). %$ . 0$ 4=+< .+ .% 4+'.&H$%( [0, 2π)<C(& %( /0. @.-(* (3'.+4)( '()$* %$* *(%0 4(+.*1K.*0-4.+)(
sen(x) =1
2=⇒ x =
π
6+ 2kπ k = 0,±1,±2, . . . =
5π
6+ 2kπ k = 0,±1,±2, . . .?% .L.-C%( -0.*'&$ /0. . 0$ 4(+.* ). %$ F(&-$ sen(x) = a E $+8%(6$-.+'. cos(x) = a<'4.+.+ F$-4%4$* 4+D+4'$* ). *(%0 4(+.*1M4 @$* F$-4%4$* (+*4*'.+ ). %$* *(%0 4(+.* 38*4 $* .+ 0+ 4+'.&H$%( ). %$&6( 2π< /0. 0*0$%-.+'..* [0, 2π)< $ %$* 0$%.* *. %.* *0-$ 0+ -,%'4C%( .+'.&( $&34'&$&4( ). 2π1 I(& %( /0. .% C&(3%.-$*. &.)0 . $ @$%%$& %$* *(%0 4(+.* 38*4 $* ). %$* . 0$ 4(+.* 4+)4 $)$*1B$ &.*(%0 4=+ ). . 0$ 4(+.* (-( %$* 4+)4 $)$* *. F$ 4%4'$ (+ %$ (+*'&0 4=+ ). 0+$ D60&$$). 0$)$< (-( %$ -(*'&$)$ .+ %$ D60&$ N91O1 P%%G H.-(* %$&$-.+'. /0.< )$)( 0+ 8+60%(
α< .% ,+4 ( ('&( 8+60%( .+ .% 4+'.&H$%( [0, 2π) /0. '4.+. 460$% H$%(& C$&$ *.+( .* C&. 4*$-.+'.(π − α)1 B( /0. -0.*'&$ /0. %$ . 0$ 4=+< sen(x) = a< −1 ≤ a ≤ 1 '4.+. )(* *(%0 4(+.*38*4 $*< α E π − α )(+). α .* 0+ 8+60%( '$% /0. −π/2 ≤ α ≤ π/21Q.-(* '$-347+ ). %$ D60&$ -.+ 4(+$)$ /0. 0$+)( sen(α) > 0 %$* *(%0 4(+.* 38*4 $* *.@$%%$+ .+ .% C&4-.& E *.60+)( 0$)&$+'.1 ;0$+)( sen(α) < 0< %$* *(%0 4(+.* *. @$%%$+ .+ .%'.& .& E 0$&'( 0$)&$+'.1
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, !........................................................................................................................................................................................................................................................................................................................................................................................
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x
y = sen(x)
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..
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...................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................
αα........................................................................................................................................................................................................................................................................................................................................................................................
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...................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................
αα
#$%&'( )*+!, -./& $.123 42 /( 2 &( $51 sen(x) = a6 |a| < 1 !"#$ $&# '()') !" a #$ $%&!'( '!") *") +#! −1 ≤ a ≤ 1, -)"&"&(. !"# %&' ()%' %(*" /! )" ! #" 12$sen(x) = a") %$1 ( 3$4#)( α *") +#! sen(α) = a 5 −π/2 ≤ α ≤ π/2,*+,!-./ $&#) 7( 3./& $51 8'$1 $8(/ 42 sen(x) = a 23 8'2 $3(921:2 arc sen(a)6 ;&23:'.:'(<(=. (1:2'$.' 21:.1 23 9&23:'( >&20-121,$ $&# '()()') !" a #$ $%&!'( '!") #")+#1!'" *") +#! −1 ≤ a ≤ 1, 6$*($ !. )"! #" 12$sen(x) = a*1!$! #$" .()# 12$ 7'1$ 17") α = arc sen(a)8 +#! !. !) %$1 ( 3$4#)( !$ !) 1$*!'9")( [−π/2, π/2] #5( .!$( !. 14#") " a,:#"$/( |a| 6= 1 *!$/'!&(. #$" .()# 12$ ;3.1 " "/1 1($")8 π − α, <(/". )". .()# 1($!. /! )"! #" 12$ !.*3$ !$*($ !. /"/". 7('
x =
{
α + 2kπ k = 0,±1,±2, . . . 2π − α + 2kπ k = 0,±1,±2, . . .*+,!-./ $&#) 7(3 (/ &/(4.'(3 %'?@ (3 8'.A221 /( 3./& $51 8'$1 $8(/ 42 /( 2 &( $51 sen(x) =
a6 924$(1:2 /( :2 /( (' 321 6 >&2 23:? :(9<$B1 ( A2 23 $421:$@ (4( .9. 321−1 +34!5261 '()()7) C(//(' :.4(3 /(3 3./& $.123 42 /( 2 &( $51 sen(x) = −1/2+
!"!" #$%&$'()#* +,'-()(./+,'$&* !"#$%$ &' &( )'*&+,-($ [−π/2, π/2] &( .') $ 0'12($ 32& *)&'& 4&'$ )12-( - −1/25 &4 −π/65 *&'&6%$4 32& arc sen(−1/2) = −π/65 7$+ ($ 32& −π/6 4&+0 '2&4*+- 4$(2 )8' 7+)' )7-(9 :'*$' &45(- $*+- 4$(2 )8' ;04) - &4 π − (−π/6) = 7π/69 <$+ ($ *-'*$5 *$=-4 (-4 4$(2 )$'&4 4$'−π
6+ 2kπ, 8 7π
6+ 2kπ, k &'*&+$ 2-(32)&+-. !"#$%& '()()() >-((-+ *$=-4 (-4 4$(2 )$'&4 =& sen(x) = 0.259?4-'=$ 2'- -( 2(-=$+- @&' (- %$=-()=-= =& +-=)-'&4A5 $;*&'&%$4 32& (- 4$(2 )8' 7+)' )7-(&4
arc sen(0.25) ≈ 0.2527B2&1$5 (- $*+- 4$(2 )8' ;04) - &4 π − 0.2527 ≈ 2.88909 <$+ ($ *-'*$5 *$=-4 (-4 4$(2 )$'&44&+0'0.2527 + 2kπ, 8 2.8890 + 2kπ. !"#$%& '()()*) >-((-+ *$=-4 (-4 4$(2 )$'&4 =& (- & 2- )8' sen(x) = 09#$%$ &' &( )'*&+,-($ [−π/2, π/2] &( .') $ 0'12($ 32& *)&'& 4&'$ )12-( - C &4 &( C5 *&'&%$432& arc sen(0) = 05 7$+ ($ 32& C &4 '2&4*+- 4$(2 )8' 7+)' )7-(9 :'*$' &45 (- $*+- 4$(2 )8';04) - 4&+0 π − 0 = π9 <$+ ($ *-'*$5 *$=-4 (-4 4$(2 )$'&4 4$'
0 + 2kπ, 8 π + 2kπ, k &'*&+$ 2-(32)&+-.D 4&-50, 2π,−2π, 4π,−4π, . . . 8 π,−π, 3π,−3π, . . .E4F 32&5 &' &4*& -4$5 7$=&%$4 &4 +);)+ *$=-4 (-4 4$(2 )$'&4 $%$ 2'- 4$(- G-%)()-5sen(x) = 0 =⇒ x = kπ, k &'*&+$ 2-(32)&+-.B- H12+- "I9J %2&4*+- (- 4)*2- )8' 7-+- & 2- )$'&4 ;04) -4 $' $4&'$9+",-. .0- '()1) !" a #$ $%&!'( '!") *") +#! −1 ≤ a ≤ 1, -)"&"&(. !"# %&' ()%' %(*" /! )" ! #" 12$
cos(x) = a") %$1 ( 3$4#)( α *") +#! cos(α) = a 5 0 ≤ α ≤ π,B- 4$(2 )8' 7+)' )7-( =& (- =&H') )8' -'*&+)$+ &4 arc cos(a)5 &4 =& )+5 &4 K&( 0'12($ @7+)' )7-(A 2L$ $4&'$ &4 )12-( - aM9 N8*&4& 32& arc cos 7+$=2 & 0'12($4 &'*+& 0 L π5 2' )'*&+,-($ =&(-+1$ π 7&+$ =)G&+&'*& -( )'*&+,-($ =$'=& &4*0' ($4 ,-($+&4 7+$=2 )=$4 7$+ arc sen9 O)12& =&(- H12+- (- 4)12)&'*& 7+$7$4) )8'9
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, !........................................................................................................................................................................................................................................................................................................................................................................................
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α
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........................................................................................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................
α
α
#$%&'( )*+,- ./0& $/234 564$ (4 73 cos(x) = a8 |a| < 1 !"#"$% %'( )*+*+,+ !" a #$ $%&!'( '!") #")+#,!'" -") +#! −1 ≤ a ≤ 1. /$-($ !0 )"! #" ,1$cos(x) = a-,!$! #$" 0()# ,1$ 2',$ ,2") α = arc cos(a)3 +#! !0 !) %$, ( 4$5#)( !$ !) ,$-!'6")( [0, π] #7( (0!$( !0 ,5#") " a.8#"$9( |a| 6= 1 -!$!&(0 #$" 0()# ,1$ :40, " "9, ,($")3 −α. ;(9"0 )"0 0()# ,($!0 9! )"! #" ,1$ !0-4$ !$-($ !0 9"9"0 2('
x =
{
α+ 2kπ, k = 0,±1,±2, . . . 1−α+ 2kπ, k = 0,±1,±2, . . . , a = 1 ( a = −13 <"7 0()# ,($!0 %$, "03 +#! 0($ )"0 0()# ,($!0 2',$ ,2")!03 = 7 π '!02! >-,6"&!$-!.-./0#1" )*+*+2+ 9(00(' :/7(4 0(4 4/0& $/234 73 cos(x) = −1/2+;( 4/0& $<2 ='$2 $=(0 73 0( 3 &( $<2 34 x = 2π/3 >8 =/' 0/ :(2:/ 0( /:'( 4/0& $<2 ='$2 $=(0 34
x = −2π/3+ ?/' 0/ :(2:/ :/7(4 0(4 4/0& $/234 4/2x = ±2π
3+ k · 2π.
!"!" #$%&$'()#* +,'-()(./+,'$&* !" !"#$%& '()()*) #$%%$& '()(* %(* +,-.%(* θ '$%/* 0./ sen(2θ) = cos(θ)1 !"#$% '()* 2(3( sen(2θ) = 2 sen(θ) cos(θ)4 */ '5/,/ 0./ %$ / .$ 57, )$)$ /* /0.58$%/,'/ $2 sen(θ) cos(θ) = cos(θ).95 cos(θ) 6= 04 %$ / .$ 57, $,'/&5(& */ *53:%5; $ $ 2 sen(θ) = 14 ( */$ $ sen(θ) = 1/21 <(& %('$,'(4 '()$* %$* *(%. 5(,/* /*'+, )$)$* :(&
θ '$%/* 0./ cos(θ) = 04 /* )/ 5&θ = ±π
2+ 2mπ, m /,'/&( $&=5'&$&5(
θ '$%/* 0./ sen(θ) = 1/24 /* )/ 5&θ =
π
6+ 2kπ, 74 θ =
5π
6+ 2kπ. !"# % %&' ()*)* ! "# %#&'(' &%)(* +(* *%+, -%#.* ). +(* *-/,-.#&.* . ,( -%#.*!0(1 sen(t) =
√3/2 021 cos(t) =
√2/20 1 sen(t) = −
√2/2 0)1 cos(t) = −
√2/20.1 tan(t) = 1 031 tan(t) = −10/1 cos(t) = 0 041 sen(t) = 15! 6(++(' &%)(* +(* *%+, -%#.* ). +(* *-/,-.#&.* . ,( -%#.*7 *,8%#-.#)% 9,. x .* ,# #:;.'% '.(+!0(1 cos(x) = −1 021 cot(x) − 1 = 00 1 2 sen(x)−
√2 = 0 0)1 sen(2x) = 1/2<! 6(++(' &%)(* +(* *%+, -%#.* ). +(* *-/,-.#&.* . ,( -%#.*! =,8%#/( 9,. x .* ,# #:;.'% '.(+ >9,. θ .* ,# ?#/,+% ;.)-)% .# /'()%*!0(1 sen2(x)− 1 = 0!021 3 cos2(x) = cos(x)!0 1 2 sen2(θ)− 3 sen(θ)− 2 = 0!0)1 2 sen2(θ)− sen(θ) − 1 = 0!0.1 2 sen(3θ) = 1!031 1 + cos(θ)
cos(θ)= 2!0/1 cos(x)−
√
cos(x) = 0!041 cos(θ)√
1 + tan2(θ) = 1!@! A'%2(' 9,. sen(x) = sen(y)7 **-7 x = y + nπ % x = (2n+ 1)π − y7 n .#&.'% ('2-&'('-%!B! A'%2(' 9,. tan(x) = tan(y)7 **-7 x = y + nπ7 n .#&.'% ('2-&'('-%!
! !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! !"#" $%&'() +,- .% /0+1-2)('&"# $%&'()(*+$%,# +- +. +-$/0&( 0+ .#- *+0&0#- 0+ .(- +.+*+)$(- 0+ /) $%&1)'/.( 2 .#- %+.# &(4)+- +)$%+ 5-$#-6 ".#*#*(- !"#$% '() *! %) + ',)-%$# # .# 0+$+%*&)# &7) 0+ +#*." .#-*+0&0#- #-( �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ρE 2 0+. ,% /.( &% /)- %&$( CRE9 +$ 6/#)0!)'#1 H) +-$+ #<,$/.( -/<()0%+*(- A/+ 0+ &% A/+ /) 1)'/.( +- ()( &0(9 +- +A/&I#4.+)$+ # #J%*#% A/+ ()( +*(- -/ -+)( 2 -/ (-+)(6 H) .# <%1 $& #9 +-$( +A/&I#.+ # 0+ &% A/+-& )(- 0#) +. 1)'/.( <(0+*(- K#..#% -/ -+)( ( (-+)( () .# #2/0# 0+ /)# #. /.#0(%# ( A/+0#0#- .#- %#;()+- $%&'()(*5$%& #- <(0+*(- K#..#% .# *+0&0# 0+. 1)'/.(6 !"#" " $%& '()*+,-.%& /0 2*+,-.%&D)& &#%+*(- )/+-$%( +-$/0&( () .(- $%&1)'/.(- *1- -&*<.+- 0+ #)#.&;#%9 .(- $%&1)'/.(- %+ 4$1)'/.(-6 @+ <%&*+%# &)$+) &7) -#G+*(- A/+ /)( 0+ -/- 1)'/.(- +- %+ $( 2 A/+ .(- ($%(- 0(-1)'/.(- -() #'/0(- 2 (*<.+*+)$#%&(- +)$%+ -,6F0+*1-9 .(- %&$+%&(- 0+ ()'%/+) &# 0+ $%&1)'/.(- %+ $1)'/.(- -+ %+0/ +) <(% .#- %+.# &()+-&)0& #0#- #)$+%&(%*+)$+6 "(- -&'/&+)$+- %+-/.$#0(- $+7%& (- )(- #2/0#%1) +) .# %+-(./ &7) 0+$%&1)'/.(-62 #3#"' '() 4516141 !" α #$ %$&#'( "&#)( #+( (,!$( !, c- !$.($ !, ,# ,!$( !, √1− c2/0$%'(&"1!$.!- ,2 ,# ,!$( !, s- ,# (,!$( !, √1− s23!1(,.4" 25$/ L#G+*(- 0+. #<,$/.( #)$+%&(% A/+ <#%# /#.A/&+% 1)'/.( #'/0( α -+ /*<.+A/+cos2(α) + sen2(α) = 1.@+ 0()0+ -&'/+) &)*+0&#$#*+)$+ .#- %+.# &()+- &)0& #0#-6
!"#" $%&'()*+,- .% /$+0-1)('& !"#$ &'& ()&* (+ &*('& , (+ *('& -( .' /'0.+& α1 23)4$5' &'& ()&* +3 23'0('2( -( α ,36.( tan(α) = sen(α)cos(α) 7 8(3)&* 39&:3 6.5 *. (-( .3'-& &'& ()&* +3 23'0('2( -( .'& -( +&*/'0.+&* 30.-&*7 !"#"$% %'( )*+,+-+ !" α #$ %$&#'( "&#)( #+" ,"$&!$,! !- t. !$,($ !-
cos(α) =1√
1 + t2sen(α) =
t√1 + t2
./!0(-,1" 23$4 ;3*23 &'*$-(:3: +3 *$2.3 $<' $+.*2:3-3 (' +3 *$0.$('2( =0.:37....................................................................................................................................................................................................................................................................................................
α
t
1
√1+t2
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................
>3:3 +3* 3?+$ 3 $&'(*1 23)4$5' :(*.+23 $)?&:23'2( (+ *$0.$('2( :(*.+23-&1 6.( '& (* )/* 6.(&2:3 )3'(:3 -( (* :$4$: +3* -(='$ $&'(* -( *('& , &*('&7 !"#"$% %'( )*+,+*+ !" T #$ ,12%$&#'( 1! ,%$&#'( ($ 526(,!$#-" h4 7$,($ !-....................................................................................................................................................................................................................................................................................................
α
A B
C
AB = AC ∗ cos(α)
BC = AC ∗ sen(α)..............................@* -( $: 6.( !' '")( ")+" !$,! " #$( )! '(- %$&#'(- "&#)(- !- 2&#"' " '" 526(,!$#-" 0#',286'2 ")" 6(1 !' (-!$( )!' %$&#'( , !' '")( (6#!-,( " #$( )! '(- %$&#'(- "&#)(- !- 2&#"' " '"526(,!$#-" 0#',26'2 ")" 6(1 !' -!$( )!' %$&#'(7 !"#"$" %&'()* ,-. /& 01,2.3*)(' %& 42.3*)('@*2.-$3:()&* +&* :$2(:$&* -( &'0:.(' $31 6.( 36.A *( :(-. (' *$)?+()('2( 3 B3-&CB3-& ,B3-&CD'0.+&7
!! !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! !" #$%!" &!'! )%!" *##+ !"#$% %#% '(%) $%) $' *%) *"$%) $' !# +,-.#/!*% ,' +.#/!*%0 %1+'#'(%) '* +', ',% ('2$-"#+' !#" "3*- " -4# $'* +'%,'(" $' 5-+./%,")0 % )'" ('$-"#+' *" ,'*" -4#a2 + b2 = c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x0 /) /567%-582 #%$ arc cos(x)! 9%$ $28%*)/ :') ;)$)6%/ 6+/ 2&)-2*1)0 /)$) %65)*&2 )- '/% &) arc cos #2$2 &)1)$65*2$ -%/ +*,'-%/ &) '* 1$5+*,'-%!<*2 ;)8 %*% 5&%/ -%/ 1$)/ -2&%/0 -%/ %1$%/ )-)6)*1%/ #')&)* 2- '-2$/) %6% /) 5*&5 2 2 %*15*'2 5=*4- #)$>6)1$% &)- 1$5+*,'-% $) 1+*,'-% )/ P = a+ b+ c . /' +$)2 (1/2)ab!4- >$ '-% 5$ '*/ $51% 15)*) 2 c %6% &5+6)1$% .0 #%$ -% 12*1%0 /' $2&5% R )/ c/2!4- >$ '-% 5*/ $51% 15)*) '* $2&5% ρ = K/s &%*&) K )/ )- +$)2 &)- 1$5+*,'-% . s )/ )-/)65#)$>6)1$% &)?*5&% %6% s := (a + b + c)/2! 4/1) )/ '* $)/'-12&% ,)*)$2-0 '.2&)6%/1$2 5=* &2$)6%/ 6+/ 2&)-2*1)! !"#$%& '()*)') @2--2$ 1%&%/ -%/ )-)6)*1%/ &) '* 1$5+*,'-% $) 1+*,'-% 12- :') /' A5#%1)B*'/2 65&) C" 6 . :') '*% &) /'/ 21)1%/ 65&) C 6! !"#$% '()* 9$56)$% &57'D26%/ '* 1$5+*,'-% :') 1)*,2 -2/ &56)*/5%*)/ )/#) 5? 2&2/0 #%B*5)*&% -%/ *%67$)/ *) )/2$5%/ #2$2 5&)*15? 2 5=*!...............................................................................................................................................................................
αA B
C
.
.
.
.
.
.
.
.
..........E'#%*)6%/ :') c = 15 . b = 10! 9%$ 951+,%$2/0 1)*)6%/ :')a =
√
c2 − b2 =√225− 100 =
√125 6.
!"#" $%&'()*+,- .% /$+0-1)('& !"# $%&% cos(α) =b
c=
10
15' ()*+,% (+ *. (.*,%/* %012+2&%) 3(2 α = arc cos(10/15) ≈
48.19◦# 4(25%' 2. %1/% 6+5(.% *5(,% &7,2 90− 48.19 = 41.81◦#8)*+,% .*) 9:/&(.*) 7+,7 *,*)' 12+2&%) 3(2;2/<&21/% = 15 + 10 +√125 = 25 +
√125 ≈ 36.18 &#=/2* =
1
2ab =
1
215 · 10 = 75 &2#>. /*,7% R ,2. </ (.% 7/ (+) /71% 2) c
2= 7.5 &#>. /*,7% ,2. </ (.% 7+) /71% 2) *?/%@7&*,*&2+12 75/(36.18/2) ≈ 4.15 &# ! "#$% & '! (!)'*% +%!% -$%. /"01$(*+,% %+% 2&%) (+% ,2 6+5(.%) *5(,%)' ,75*&%) α' 2. %1/% 6+5(.% 2) 75(*. *. %&?.2&2+1%,2. ?/7&2/%# A( &2,7,* 2) 75(*. * (90− α)◦#B2+2&%) .%) )75(72+12) *)%)# A7 2. .*,% %+% 7,% 9(2/* .* C7?%12+()*' ()*+,% .* ?/%?%)7 7:+DE# #E %012+2&%) .%) ,%) *121%)# 8+* F2G %+% 7,% .%) 1/2) .*,%)' ?/% 2,2&%) * ,212/&7+*/.*) &2,7,*) ,2 .%) %1/%) 2.2&2+1%) %&% 2+ 2. *)% *+12/7%/#A7 2. .*,% %+% 7,% 2) (+% ,2 .%) *121%)' ,75*&%) b' H 2. 6+5(.% 2) α' %&% )2 7+,7 * 2+ .*)75(72+12 I5(/*'
...............................................................................................................................................................................
αA B
C
.
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.
..........12+2&%) 3(2 %&%a
b= tan(α)?%,2&%) *. (.*/ a# $%+% 7,%) .%) *121%)' ()*&%) ;7165%/*) ?*/* ,212/&7+*/ .* C7?%12+(J)* c# 4(25%' +(2F*&2+12' 12+2&%) %+% 7,%) .%) 1/2) .*,%) H ?/% 2,2&%) %&% 2+ 2. *)%*+12/7%/#23456*% 789:9;9 K*..*/ 1%,%) .%) 2.2&2+1%) ,2 (+ 1/76+5(.% /2 16+5(.%' )*072+,% 3(2 )(C7?%12+()* &7,2 LM H 3(2 (+% ,2 )() 6+5(.%) *5(,%) &7,2 E"#M ◦# !"#$% '()*
...............................................................................................................................................................................
α
β
A B
C
25
.
.
.
.
.
.
.
.
..........A(?%+72+,% 3(2 2. 6+5(.% ,*,% 2) α' 12+2&%) 7+&2,7*1*&2+12 3(2 2. %1/% 6+5(.% &7,2 ML#M◦#
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, !b = c cos(α) = 25 cos(37.5) ≈ 19.83.
a = c sen(α) = 25 sen(37.5) ≈ 15.22.#$%&' (' )* +,-'.$*. /&'0 1'.23'-.* ≈ 60.054 5.'+ ≈ 150.94 .+($* 2. &)* $. &,7 .$-* 8 9:;<4.+($* 2. &)* $,7 .$-* ≈ 5.03; !"# % %&' ()*+* ! "# $%&'()*+,# -) +* (.%/*0+1' .) (/*0+1' 3%-) 45 &+10#-#, 6 +*' -) ,+, /*0+1', #0+-', 3%-)47 0.#-',! 8#11#. 1# 3)-%-# -) 1', '(.', -', 1#-',!9! "', #()(', -) +* (.%/*0+1' .) (/*0+1' 3%-)* : 6 5! 8#11#. 1# $%&'()*+,#; 1', /*0+1', 6 )1/.)# -)1 (.%/*0+1'!<! "', #()(', -) +* (.%/*0+1' .) (/*0+1' 3%-)* : 6 97! 8#11#. )1 /.)# -)1 (.%/*0+1' 6 1', .#-%',-) 1', =. +1', %*, .%(', 6 %. +*, .%(',!>! ?% 1', .#6', -)1 ,'1 @'.3#* +* /*0+1' -) 9A◦ '* )1 ,+)1'; B +/*(' 3)-%./ 1# ,'3C.# -) +*#
# +6# #1(+.# ), <5 &%),D4! E* %).(' 3'3)*('; 1', .#6', -)1 ,'1 @'.3#* +* /*0+1' -) 9<◦ '* )1 ,+)1'! BF+/* #1(' ), +*3/,(%1 -) C#*-).#; ,% ,+ ,'3C.# )* ),) %*,(#*() 3%-) <4!7 &%),D7! GF'*(%*+# %H*!I BF+/1 -)C).=# ,). )1 /*0+1' -) )1)J# %H* G)1 /*0+1' K+) 1', .#6', ,'1#.), @'.3#* '* )1 ,+)1'I &#.# K+) 1# ,'3C.# @+).# )1 -'C1) -) 1#.0'D:! L &.'&H,%(' -) ,'3C.#,; B +/*-' +* &',() &.'-+ %./ 1# ,'3C.# -) 3#6'. 1#.0'D BM'-.=#,+ )-). K+) +* &',() *' &.'-+N).# ,'3C.# #10+*#D B +/*-'D B-H*-)DA! O* ), #1).# -) 4 &%), -) 1#.0' ),(/ #&'6#-# )* +*# &#.)- @'.3#*-' +* /*0+1' -) :5◦ '* )1,+)1'; B# K+P #1(+.# -) 1# &#.)- ),(/ #&'6#-# 1# '(.# &+*(# -) 1# ), #1).#DQ! G#I O* /*0+1' )*(.#1 θ -) +* =. +1' G+* /*0+1' '* JP.(% ) )* )1 )*(.'I ,+C(%)*-) +*# +).-# -) 1#.0' L! ER&.),#. L )* (P.3%*', -)1 .#-%' -)1 =. +1' 6 )1 /*0+1' θ!GCI BF+/1 ), )1 &).=3)(.' -) +* ' (/0'*' .)0+1#. %*, .%(' )* +* =. +1' -) .#-%' 5DG I BF+/1 ), )1 &).=3)(.' -) +* &'1=0'*' .)0+1#. -) 55 1#-', %*, .%(' )* +* =. +1' +*%(#.%'DBM'. K+P ,+ .),&+),(# -)C).=# ,). #,% '3' 2πD 5! M.'C#. K+) )* +* (.%/*0+1' .) (/*0+1' +6# $%&'()*+,# 3%-# c 6 ()*0# +* /*0+1' '*' %-' α;)1 /.)# &+)-) '3&+(#.,) +,#*-' 1# @H.3+1#S =
1
2c2 cos(α) sen(α).G#I (1, 1)! GCI (1, 5)! G I (7, 12)! G-I (3.75, 2.78!
!"#" $%&%' (% '%)* & +*'%)* !" ! "#$%&' (%$ $* (*$&'+$, &-.-+/0%$+1$2 arc sen(sen(x)) 3$+$ x = 10◦, 25◦, 50◦!142 sen(arc sen(x)) 3$+$ x = 0.1, 0.5, 1.0, 1.5! !"#" $%&%' (% )%*+ & ,+'%*+#$ %&'% ()*+, -&.+%/,& (. $,'. )1$ )$2) .2. %$ (. &)3-)%$'% 43-+. 5.+. +%6%+)+$,& . '+)7$3-(,& -.(%&8-)%+.9.............................................................................................................................................................................................................................................................................................................................................................................................................................
α β
γ
A B
C
ab
c:)3-+. ;<9=> ?+)7$3-(, -.(8-)%+.@2%/7& '%$2+%/,& (,& &)3-)%$'%& ,$A%$),& $,'. ),$.(%&B 8-% '+.2) ),$.(/%$'% C.$ &)2, -&.D2,& %$ %&'% '%/.9#( 7+%. 2%( '+)7$3-(, &%+7 &)/*,()E.2. 5,+ K9%( 5%+F/%'+, 2%( '+)7$3-(, &%+7 2sB 2,$2% s = (a+ b+ c)/2 %& %( !"#$!%&"!'%()#( +.2), 2%( F+ -(, )+ -$& +)', &%+7 R9#( +.2), 2%( F+ -(, )$& +)', &%+7 ρ9G*&%+A%/,& %$ 5+)/%+ (-3.+ 8-%B ,/, (. &-/. 2% (.& /%2)2.& 2% (,& '+%& 7$3-(,& )$'%+),+%&2% -$ '+)7$3-(, %& )3-.( . ;=H◦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
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! !"#" " $%& '% ()*%+)*#$ !" #! %&!'%& %& '($ )%(%*$+,-$ ,/( 0%+ 1%2*%3$ 0% 4,15)2*$& $ &,1'$ ,2(%& 02(0% (21%(%32& '( 1*,5()'+2 *% 15()'+26 72&1*$*%32& 232 $+ '+$* a %( 18*3,(2& 0% b9 c : 0%+5()'+2 2;'%&12 $ a9 2 &%$ 0% α6 </1%&% ='% α %& %+ 5()'+2 %(1*% b : c6...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
α
A B
C
D
ab
................
c=AB
>,)'*$ ?@6AB #$ #%: 0%+ C2&%(2#$ ;*,3%*$ D)'*$ *%;*%&%(1$ %+ ;*2E+%3$ ='% 0%&%$32& *%&2+F%*6 G( +$ &%)'(0$ D)'*$ H%32&0,F,0,02 %+ 1*,5()'+2 2*,),($+ %( 02& 1*,5()'+2& *% 15()'+2&6 G&12 (2& ;%*3,1,*5 2(&1*',* +$+%: 0% 2&%(2&6 C232 &';2(%32& 2(2 ,02 %+ 5()'+2 α9 2E&%*F$(02 +$ &%)'(0$ D)'*$ 1%(%32&='%a2 = d2 + (c− n)2 I?@6JKL0%35& ;20%32& F%* ='% d = b sen(α) : ='% n = b cos(α)6 M'&1,1':%(02 %( +$ % '$ ,/( ?@6J1%(%32& ='%
a2 = d2 + (c− n)2
= (b sen(α))2 + (c− b cos(α))2
= b2 sen2(α) + c2 − 2bc cos(α) + b2 cos2(α)
= b2(sen2(α) + cos2(α)) + c2 − 2bc cos(α)
= b2 + c2 − 2bc cos(α)N%&'+1$ ='% a2 = b2 + c2 − 2bc cos(α)6 O(1%* $3E,$(02 +2& 5()'+2&9 2E1%(%32& *%&'+1$02&$(5+2)2& ;$*$ b2 : c2 (!"!& #! )%&!'%a2 = b2 + c2 − 2bc cos(α)b2 = c2 + a2 − 2ca cos(β)c2 = a2 + b2 − 2ab cos(γ)
!"#" $%&%' (% '%)* & +*'%)* !"#$ %&'&()&*+ ,&,-* ,-* '&,-* ,. /$ 0)12$3/'- 4 .' 2$3/'- .$0). .''-*+ .' /&,)&,- ,.' 0.) .)'&,- .* 13/&' & '& */6& ,. '-* /&,)&,-* ,. '-* '&,-* -$- 1,-* 6.$-* .' ,-('. ,.' %)-,/ 0-,. '-* ,-* '&,-* 4 .' -*.$- ,.' 2$3/'- 7/. 8-)6&$9 !"#$%& '()*)') :&''&) c+ *&(1.$,- 7/. a = 15+ b = 20 4 γ = 50◦9;%'1 &$,- '& '.4 ,.' -*.$- %&)& c+ 0.$.6-* 7/.c2 = 152 + 202 − 2 · 15 · 20 cos(50◦) ≈ 239.33<. ,-$,.+ c ≈ 15.479 !"#$%& '()*)+) =$ 0)12$3/'- 01.$. '&,-* a = 5+ b = 7 4 c = 89 :&''&) .' 2$3/'- α9 !"#$% '()* =*&$,- '& ).'& 1>$ %&)& a2+ 0.$.6-* 7/.
a2 = b2 + c2 − 2bc cos(α).<. ,-$,.+cos(α) =
b2 + c2 − a2
2bc.?/*010/4.$,- '-* @&'-).* ,&,-* -(0.$.6-* 7/. cos(α) ≈ 0.78579 A-$ '& &4/,& ,. /$& &' /B'&,-)&+ -(0.$.6-* 7/. α ≈ 38.21◦9#' .C.6%'- "D9E9F 6/.*0)& >6-+ ,&,- '-* 0).* '&,-* ,. /$ 0)12$3/'-+ %-,.6-* G&''&) '&*6.,1,&* ,. '-* 2$3/'-* ,.' 0)12$3/'-9 #*0- 6/.*0)& .' 8/$ 1-$&61.$0- .$ .*0. -$0.H0- ,.'0.-).6& III ,. -$3)/.$ 1&9#' .C.6%'- "D9E9" 6/.*0)& 7/. ,&,-* ,-* '&,-* 4 .' 2$3/'- 1$ '/1,- JI;IK+ %-,.6-* &' /'&).' 0.) .) '&,- 4+ %-) '- &$0.)1-)+ 0.$.6-* '-* -0)-* 2$3/'-*9 !"#"$" %&' (& )&*+,I& ).'& 1>$ ''&6&,& +!, -! $#" .!)#" $-* %.)6101)2 ).*-'@.) .' 0)12$3/'- /&$,- -$-L &6-*/$ '&,- 4 '-* 2$3/'-* &,4& .$0.* J;I;K9
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...............................................................................................................................................................................................................................................................
α β
A B
C
D
ab
................
∠C = γ
M13/)& "D9"NO I& I.4 ,. ?.$-*
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, !#$%&%'()*+ &(, -%,-(, ./01(, 203 0,(-+, 4(1( &( &35 *3 +,3)+,7 431+ )+-81()*+ &+, 9)/0&+,α7 β 5 γ $3)3-+, 203
d = b sen(α) 5 203 d = a sen(β):+-+ &+, &(*+, %'20%31*+, *3 (-8(, 3 0( %+)3, ,+) %/0(&3,7 4+*3-+, %/0(&(1 &+, &(*+, *313; <+, 4(1( +8$3)31 203b sen(α) = a sen(β). =>?@!AB3 *+)*37 *%C%*%3)*+ 4+1 ab 3) (-8+, &(*+, +8$3)3-+, 203sen(α)
a=
sen(β)
b.#,()*+ &( (&$01( +113,4+)*%3)$3 ( A7 +8$3)3-+, 203
sen(β)
b=
sen(γ)
c.:+&+ ()*+ &+, 13,0&$(*+, ()$31%+13, D0)$+,7 $3)3-+, &( ,%/0%3)$3 13&( %E) !" #! $!%&'
sen(α)
a=
sen(β)
b=
sen(γ)
c.()!*+,& -./0/./ F3,+&C31 3& $1%9)/0&+ T $(& 203 c = 207 α = 50◦ 5 β = 60◦@ !"#$% '()* !"#$ !%&$'!()(&$%)$ '$ *+, '()+, -#$
γ = 180− α− β = 180− 50− 60 = 70◦..,(/$&+, *( 0$1 '$ $%+, 2(/( (* #*(/ *+, 4(*+/$, '$ a 1 b5 6+&+a
sen(α)=
c
sen(γ))$%$&+, -#$a =
sen(α)
sen(γ)c ≈ 0.7660
0.9397· 20 ≈ 16.30.7%8*+"(&$%)$9
b =sen(β)
sen(γ)c ≈ 0.8660
0.9397· 20 ≈ 18.43.
!"#" $%&%' (% '%)* & +*'%)* !" !"#$%& '()*)+) #$%&'($) $' *)+,-./'& T *0' 1/$ c = 202 α = 110◦ 3 β = 30◦4 !"#$% '()* !"#$ !%&$'!()(&$%)$ '$ *+, '()+, -#$γ = 180− α− β = 180− 110− 30 = 40◦..,(/$&+, *( 0$1 '$ $%+, 2(/( (* #*(/ *+, 4(*+/$, '$ a 1 b5 6+&+
a
sen(α)=
c
sen(γ))$%$&+, -#$a =
sen(α)
sen(γ)c ≈ 0.8397
0.6428· 20 ≈ 29.24.7%8*+"(&$%)$9
b =sen(β)
sen(γ)c ≈ 0.5
0.6428· 20 ≈ 15.56 !", . .&/ '()*):5 ;(**(/ *( &$'!'( '$ *+, 8%"#*+, 1 '$* *('+ -#$ <(*)(%=(> a = 109 b = 189 γ = 78◦5=?> b = 31.19 c = 69.19 α = 79◦5= > c = 80.89 a = 71.49 β = 1075@5 ;(**(/ *( &$'!'( '$* 8%"#*+ 1 '$ *+, *('+, -#$ <(*)(%5=(> c = 17.39 α = 52◦9 β = 65◦5=?> a = 59.19 β = 66◦9 γ = 86◦5= > b = 68.29 γ = 88◦9 α = 64◦5A5 ;(**(/ $* 2$/B&$)/+ 1 $* 8/$( '$* )/!8%"#*+ #1+, '()+, ,$ !%'! (%5=(> a = 52.89 b = 38.49 γ = 104◦5=?> c = 2.49 α = β = 82◦5= > b = 64.19 γ = 61◦9 α = 56◦5='> b = 109 c = 89 α = 1005=$> c = 80.69 a = 549 β = 125.615=<> b = 65.29 γ = 59.6◦9 α = 55.25C5 .% (4!D% 4!(E( FG &!**(, H( B( $* $,)$ '$,'$ ,# 2#%)+ '$ 2(/)!'(9 *#$"+ /+)( :G◦ H( !( $* %+/)$1 4#$*( :GG &!**(, ('! !+%(*$, H(,)( ,# '$,)!%+5 I6#8* $, *( '!,)(% !( '$,'$ $* 2#%)+ '$ 2(/)!'(H(,)( $* '$,)!%+J
! !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! ! "#$ %& '#$ '(%#$ (%)( &+,&$ %& -+ .(/('&'#0/(1# %&,&/12+(+ -+ 3+0-'# %& ◦ ) 12%&+ 4 ) 1! 56-3+,# 12%&+ '($ %2(0#+('&$ %&' .(/('&'#0/(1#78! "#$ 9#,&$ .(/,&+ %& -+ .-+,# #1:+ &+ ,/()& ,#/2($ /& ,2';+&($ <-& =#/1(+ &+,/& $; -+ 3+0-'#%& 8◦! >2 '( /(.2%&? %&' ./21&/# &$ @A .2&$B$&0 ) '( %&' $&0-+%# @ .2&$ .#/ $&0-+%#C 5 -3'$&/3 '( %2$,(+ 2( &+,/& &''#$ %&$.-D$ %& @!4 12+-,#$7E! F+ -+ ,/23+0-'# GH6C α = 70◦C b = 12! I(''(/ '( 1&%2%( %& '( (',-/( $#9/& &' '(%# c!4! J$(/ '#$ %(,#$ .(/( #1.-,(/ '( 1&%2%( %& '#$ '(%#$ ) 3+0-'#$ =(',(+,&$C ($; #1# D' 3/&( %&',/23+0-'#!K(L α = 20◦C β = 50◦C c = 12!K9L a = 10C b = 12C γ = 110◦!K L a = 5C b = 6C c = 10!M! J$(+%# -+ (/0-1&+,# 0D,/2 # #1# (<-&' -$(%# &+ &' ,&N,#C ./-&9& <-& .(/( -('<-2&/3+0-'# (0-%# α $& -1.'& <-&cos(90 + α) = − sen(α) sen(90 + α) = cos(α).OA! P&/2Q (/ <-& -(+%# &' ,/23+0-'# &$ /& ,3+0-'#C '( R&) %&' 6#$&+# #//&$.#+%2&+,& (' 3+0-'#/& ,# &$ '( /&'( 2S+ %&' T&#/&1( %& U2,30#/($!OO! 56-3'&$ $#+ '#$ .#$29'&$ V('#/&$ %& cos(α)C sen(α)C tan(α)C .(/( 0◦ ≤ α ≤ 180◦7O@! 56-3'&$ %& '($ $20-2&+,&$ 2%&+,2%(%&$ &$ 1&W#/7cos(α) + sen(α) < 2 S cos(α) + sen(α) ≤ 2.OX! U/#9(/ <-& &' ,/23+0-'# T = △(ABC) ,2&+& -+ 3+0-'# #9,-$# &+ BC $$2C c < b cos(α)!OY! U/#9(/ <-& &' ,/23+0-'# T = △(ABC) ,2&+& -+ 3+0-'# #9,-$# &+ CC $$2C c > b/ cos(α)! !"#" $%&'( )*+,+-%'( .+* /&01-23*'"#$ $% '()%$ #)*%+'(+%$ ,-%$*+#) (,( (.*%)%+ /($ *+%$ /#0($ 1 *+%$ 2)3-/($ ')*%+'(+%$ 0% -)*+'2)3-/(4 -#)0( $% ()( %) /($ *+%$ /#0($ 5"""67 ( 0($ /#0($ 1 %/ 2)3-/( ') /-'0( 5"8"67( -) /#0( 1 /($ 2)3-/($ #01# %)*%$ 58"869 :%+%,($ #;(+# (,( <(0%,($ #/ -/#+ (*+($%/%,%)*($ ( #)*'0#0%$ +%/# '()#0#$ () %/ *+'2)3-/(9
!"#" $%&$' ()(*(+%$' ,() %&-.+/0)$ !" ! "#$% &$! '#()*+,!-#$% '()*+$% ),-, .,//,- /, #01 20 /$% 304$% (*0%+-,4 5*0 0/ 6-0, K )*020 ,/ */,-%076 8/(04+0 04 0/ ,%$ #9#: $($K =
1
2bc sen(α) =
1
2ca sen(β) =
1
2ab sen(γ).928 8$4,/(04+0: /, -0/, 8'4 ,4+0-8$- (*0%+-, 0/ ;,/$- 20 /, -,<'4 04 /, #01 20 304$%: 1, 5*0(*/+8)/8 ,42$ )$- 2/abc $=+040($% 5*0
2K
abc=
sen(α)
a=
sen(β)
b=
sen(γ)
c. ! .%&(- &$! /0# ,!- 2*3 #(4-#, 5(3$ 4#(6 20 *4 64>*/$ 0% *4, /?40, 5*0 +8040 /, %8>*804+0 )-$)802,2 !"! #$%&' "( )! *+,( &.+/ "( $% 0%1$)' (2$+"+,&! "( )', )!"', "() 0%1$)' 3 !"!#$%&' (2$+"+,&!%&( "( )', )!"', "() 0%1$)' (,&0 ,'*.( )! *+,( &.+/4@%+$ 8()/8 , 5*0 *4, 8- *470-04 8, $4 04+-$ 04 *4 )*4+$ 20 /, =8%0 +-8< 1 -,28$ /, 28%+,4 8,20/ 04+-$ , *4$ 20 /$% /,2$%: 0% +,4>04+0 , ,(=$% /,2$% 20/ 64>*/$A
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s. !"#"!" $%&'()* +, )* -*./,.0, +,) 1./()2 3,+42DG.('%-+(2+ 6+0+-$, 4(' ;@0-4%' /'0' %' 2'(F+(2+ )+ %' -.2') )+ 4($ )+ %$, H(F4%$,D I/0$=6+ ?'0+-$, %' 8>,34+)' )+ +,2', ;@0-4%',* /'0' -$,20'0 +% ,.F(.J ')$ F+$-720. $ )+ %',+</0+,.$(+, s− a* s− b 9 s− cD#$(,.)70+,+ %' ,.F4.+(2+ JF40'DK( %' JF40'* x +, %' ).,2'( .' )+,)+ +% 6702. + A ?',2' +% /4(2$ )+ 2'(F+( .' )+ %' .0 4(=;+0+( .' .(, 0.2'* 9 +,' ).,2'( .' +, $->( /'0' '-8$, %')$, )+% H(F4%$ αD LF4'% ,.F(.J ')$2.+(+( y 9 z $( 0+,/+ 2$ ' %$, H(F4%$, β 9 γDM@2+,+ 34+ 2s = 2x + 2y + 2z = 2(x + y + z) 9 34+ /$0 %$ 2'(2$* s = x + y + zD N4+F$*
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.E,<-/' "C7"DF G,*.$1' #$ ))/#$('#'* &)%'/$*?)* &-(.)* #$ -(' 1,*1' ,/ -(H$/$( ,' $*.;( '/' .$/,I'#)* &)/ .$($/ .)#)* *- &/,1$/' ))/#$('#' &)%'/ ,<-'% '% /'#,) #$ %' ,/ -(H$/$( ,'6 1,$(./'* @-$ %)* /'4)* @-$ *'%$( #$% )/,J<$( *)( &-(.)* -4' *$<-(#' ))/#$('#'6 $% ;(<-%)6 $* )(*.'(.$7 K)1) %)* %'#)* .$/1,('%$*#$ %)* ;(<-%)* ),( ,#$( '#' 2π6 L'4 @-$ *$%$ ,)('/ -( ,(.$/5'%) #)(#$ L'4' -(, ,#'#7 ?'$%$ ,9( 1;* )10( $* (−π, π]7M'/' $H$ .)* #$ ))/#$('#'* &)%'/$* &)#$1)* &$(*'/ @-$ '#' &-(.) #$% &%'() #$.$/1,('-( 5$ .)/6 ) *$' -(' N$ L' @-$ 5' #$*#$ $% )/,<$( '% &-(.)6 %' #,*.'( ,' r #$% &-(.) '% )/,<$($* &/$ ,*'1$(.$ $% %'/<) #$ %' N$ L'7 G, $% &-(.) .-5,$/' )1) ))/#$('#'* '/.$*,'('* $%&'/ (a, b)6 25$/ %' B<-/' "C7"O $(.)( $* .$(#/P'1)* @-$r =
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x
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..............
r
θ
(a,b)
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.E,<-/' "C7"OF&)/ )./) %'#)6 #'#' -(' #,*.'( ,' r 4 -( ;(<-%) θ &)#$1)* /$ -&$/'/ $% &-(.)6 4' @-$ '%
!" !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, !#$%&' & ()& *+$%&) +& r *#- .'+/#)0 *+ 1. 2()%. #$%3 $.4'# ()& +' ()5#'#) +& *# '&*+. r .) #)%'. #) #- .'+/#)6 7(#/. .) #- 3)/(-. *#%#'8+)&8.$ #- 2()%. $.4'# -& +' ()5#'#) +&6 !"#" " $%&' )*+%, %+-.% &*, /),-%0', 1% 2**.1%+'1',9(2.)/&8.$ :(# %#)#8.$ () 2()%. *# ..'*#)&*&$ &'%#$+&)&$ (x, y) ; *# ..'*#)&*&$2.-&'#$ (r, θ)6 <)%.) #$ $# %+#)#0 =#' -& >/('& ?@6?A0 :(#x = r cos(θ) ; y = r sen(θ).<$&$ '#-& +.)#$ ).$ 2#'8+%#) .4%#)#' -&$ ..'*#)&*&$ &'%#$+&)&$ .). +#)*. -&$ ..'*#)&*&$2.-&'#$6 B&'& +' #) $#)%+*. .)%'&'+.0 .4$#'=#8.$ :(# x2 + y2 = r2 cos2(θ) + r2 sen2(θ) = r2; :(# y/x = tan(θ)6 C $#&0 :(# .). +*&$ -&$ ..'*#)&*&$ (x, y) 2.*'#8.$ 1&--&' r ; θ 2.'-&$ '#-& +.)#$r =
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xA = 5cos(60◦) =5
2, yA = 5 sen(60◦) =
5√3
26"6 B = (7, 3 '&*+&)#$ )
xB = 7cos(3) ≈ −6.92990 yB = 7 sen(3) ≈ 0.98786 !"#$%& '()*)+) D&--&' -&$ ..'*#)&*&$ 2.-&'#$ *# -.$ 2()%.$ (;&$ ..'*#)&*&$ &'%#$+&E)&$ $# +)*+ &)6?6 A = (5, 3)
rA =√52 + 32 =
√340 tan(θA) = 3
5=⇒ θA ≈ 0.2498 '&*+&)#$6"6 B = (−7,−5)
rB =√72 + 52 =
√746 B.' $( 2&'%# tan(θ) =
−7−5 6 7.$ $+/).$ *# -&$ ..'*#)&*&$ ).$+)*+ &) :(# #- 2()%. #$%3 #) #- %#' #' (&*'&)%#6 7(#/.
θB = −π + arctan(7/5) = −2.1910.
!"#" $%%&'()*'*+ ,%-*&(+ !" !"#"$" % '( )*+,- ,+ .**/0,+(0(- 1*2(/,- !"#$ %& ()*+,-#$ (&+%.!#.%$ #"# ($# !(. /0,.($ #& .%-( !#&%$ (-0%1.(! ($2 3,% --("("#$-($ % ,( !#&%$ 4% -( /0,.(5 6,(&4# ,$("#$ ##.4%&(4($ )#-(.%$2 +("1!7& )#4%"#$ #1+%&%.% ,( !#&%$ 4% -($ /0,.($2 3,% 0%&%.(-"%&+% +!%&%& ,&( 8#."( ",9 4!$+!&+( ( -($ % ,( !#&%$ (.+%$!(&($5 :#. %;%")-#2 %& ##.4%&(4($ )#-(.%$ r = 5 %$ -( % ,( !<& 4% ,&( !. ,&8%.%& !(4% .(4!# = 9 %&+.# %& %- #.!0%&5 !"#$%& '()*)() >(--(. -( % ,( !<& (.+%$!(&( 4% -( /0,.( #& % ,( !<& )#-(. 4(4( )#.r = 10 cos(θ)5 !"#$% '()* r = 10 cos(θ) =⇒ r2 = 10r cos(θ)5 :#. -# +(&+#2 )($(&4# ( ##.4%&(4($.% +(&0,-(.%$ +%&%"#$ 3,%
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A
θϕ
P
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√2, 300◦)! 1)2 (2, 3
4π)6! "#$$#% $# * /# -7+ *+ ((%)*+#)#& ,($#%*& )*
! !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! !" #! #$%&! '(& )!*! )+, (0, 0) - (3, (3/4)π). /" #! #$%&! '(& )!*! )+, (0, 0) - (−3, 5). " #! #$%&! '(& )!*! )+, (1, 0) - (0, 1). 1" #! #$%&! '(& )!*! )+, (3, 5) - (−2,−4). &" #! #$%&! ax+ by = c. 2" #! 3, (%2&,&% 3! +% &%4,+ &% &# +,35&% - '(& )!*! )+, (−5, 12).6. 7,!%*2+,8!, #!* & (! 3+%&* !,4&*3!%!* *35(3&%4&* ! & (! 3+%&* )+#!,&*. !" x2 + y2 = a2. /" y = (√2/2)x.9. 7,!:!, #!* 5,;< !* 1& #!* *35(3&%4&* (,=!*. !" r = 3/2. /" θ = π/4. " r = sen(θ). r = θ.>. 7,!:!, #!* 5,;< !* 1& #!* *35(3&%4&* (,=!*? &% (%! !# (#!1+,! 5,;< ! !" r = 2 + 2 cos(θ). /" 8 sen(2θ). " r = 8 cos(5θ). r = 0.2θ? 0 ≤ θ ≤ 30π.@. A,+/!, '(& #! 13*4!% 3! 1& P1 = (r1, θ1) ! P2 = (r2, θ2) &*
d(P1, P2) =√
r21 + r22 − r1r2 cos(θ1 − θ2). !"#" $%&'%(%)*+ -.) /'-01)123*'- + 4% 562%'1( 712&8%91("#$ $%&' $ ')# '#*+,+-$#*+ .+ &$ ,+%,+-+#*$ ')# %/&$, +- 0-$,&$ /# &/- #12+,/- /2%&+3/-45$6+2/- 70+ $.$ #12+,/ /2%&+3/ z = a + bi %0+.+ -+, ,+%,+-+#*$./ %/, +& %0#*/ /# //,.+#$.$- $,*+-'$#$- (a, b)4 8$- //,.+#$.$- %/&$,+- (r, θ) .+& %0#*/ +-*9# ,+&$ '/#$.$- /# +& /2%&+3/ /,,+-%/#.'+#*+ .+ &$ -':0'+#*+ ;/,2$r =
√
a2 + b2 /,,+-%/#.+ $ &/ 70+ -+ &&$2$ +& 2).0&/ / &$,:/ .+ z4 <& 9#:0&/ θ -+ .+*+,2'#$ .+ 2$#+,$$#9&/:$ $ &$ ,+%,+-+#*$ ')# %/&$,= %+,/ +# +& $-/ .+ &/- /2%&+3/- -+ &&$2$ +& !"#$%&'(.+& #12+,/ > -+ -'26/&'?$ %/, 4 @0$#./ +& /2%&+3/ z *'+#+ 2).0&/ r > $,:02+#*/ θ= *+#+2/-70+z = a+ bi = r cos(θ) + ir sen(θ) = r(cos(θ) + i sen(θ).A+ /,.$2/- &$ ,+%,+-+#*$ ')# $#*+,'/, +- ,'6'+#./
z = r∠θ= / *$26'B# r *+(θ).8++2/- &$ +C%,+-')# z = r∠θ /2/ +& /2%&+3/ r 9#:0&/DEθF= / -+$ +& /2%&+3/ 0>/ &$,:/+- r > 0>/ $,:02+#*/ +- θ4 G/, -0 %$,*+= !"(θ) +- 0#$ $6,+H'$ ')# .+ cos(θ) + i sen(θ)4<& -I26/&/ r∠θ +- 1*'& %$,$ /2%0*$, +& %,/.0 */ .+ ./- #12+,/- /2%&+3/-4 J+$2/- %/,70B4 50%/#:$2/- 70+ *+#+2/- ./- /2%&+3/-= z1 = r1∠θ1 > z2 = r2∠θ24 K -+$= z1 =
!"#" $%&$%'%()*+,-( )$,./(/01)$,+* 2% (30%$/' +/0&4%5/' !"r1(cos(θ1) + i sen(θ1) # z2 = r2(cos(θ2) + i sen(θ2)$ %&'(& *+ *, -.(/0 '( /* z1 (& z2 +*-0*/* 1, 0,1. (2(
z1z2 = r1(cos(θ1) + i sen(θ1)) · r2(cos(θ2) + i sen(θ2))
= r1r2(cos(θ1) + i sen(θ1)) · (cos(θ2) + i sen(θ2))
= r1r2(cos(θ1) cos(θ2) + i cos(θ1) sen(θ2) + i sen(θ1) cos(θ2) + i2 sen(θ1) sen(θ2))
= r1r2((cos(θ1) cos(θ2)− sen(θ1) sen(θ2)) + i(sen(θ1) cos(θ2) + cos(θ1) sen(θ2)))3(. ,1+ 4/*&'4/1/*+ /* ,1+ 50& 4(&*+ '.46(&(27'.4 1+ /* ,1+ +021 /* 1.602*&'(+8 '*&*2(+90*cos(θ1) cos(θ2)− sen(θ1) sen(θ2) = cos(θ1 + θ2)
sen(θ1) cos(θ2) + cos(θ1) sen(θ2) = sen(θ1 + θ2): +*1 90* *, 2;/0,( /* 0& -.(/0 '( /* /(+ (2-,*<(+ *+ 4601, 1, -.(/0 '( /* ,(+ 2;/0,(+8# *, 1.602*&'( /*, -.(/0 '( *+ 4601, 1 ,1 +021 /* ,(+ 1.602*&'(+$ %+* .*+0,'1/( +* ,,121 ,1 !"# %&' (! )*%+ ! =*,1 4;& /* >(4?.*(r1∠θ1)(r2∠θ2) = r1r2∠(θ1 + θ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= rk∠(kθ) · r∠θ= rk+1
∠((k + 1)θ).-./0#1" )*+,+)+ B*11*. z10 )*&2% z = 1 + i6 C1*.*/-&'-9 z =√2∠45◦6 <%. 1% '*&'%9
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z2 = (cos(θ) + i sen(θ))2 = cos2(θ)− sen2(θ) + i · 2 sen(θ) cos(θ)F0*&2% 1* .-1* (+& 2- 3%(,.-9 '-&-/%09 4%. %'.* 4*.'-9 =)-z2 = (∠θ)2 = ∠(2θ) = cos(2θ) + i sen(2θ).<%. 1% '*&'%9 (;)*1*&2% 4*.'-0 .-*1-0 - (/*;(&*.(* %E'-&2.-/%0 =)-
cos(2θ) = cos2(θ)− sen2(θ), G sen(2θ) = 2 sen(θ) cos(θ).H- )4-.*&2% 2- -0'* /*&-.* &)-0'.* 8+./)1*0 4*.* 1* 0)/* 2- :&;)1%06
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a∠04 5&6 -& /.'/& /)')0&1 2$) rn = a + 2$) nθ = 04 7. 6)-. (8' rn = a (0%-( . 2$) r = a1/n45&6 &/6. %.6/)9 + 6) &6*.'*& 2$) 0 = 2π &0& :';$-&19 /)')0&1 2$) nθ = 0 = 2π (0%-( .2$) θ = 2π/n4 5&6 -& 2$) !" *) -.1 1&-$ (&')1 1)6: r(1/n)∠(2π/n)4 <&0& 1.=)0&1 2$) $'.) $. (8' %&-('80( . *) ;6.*& n /()') )>. /.0)'/) n 1&-$ (&')19 '&1 %6);$'/.0&1 %&6 -.1&/6.1 1&-$ (&')14 7. 6)1%$)1/. )1/: &'/)'(*. )' -. 1(;$()'/) %6&%&1( (8'4#$%&%'( (*! +,-.-/ ?7. ) $. (8' zn = a@- !" a #$ $%&!'( '!") *(+,-,.(/ n #$ $%&!'($"-#'")0 ζ = ∠(2π/n)01"2 13"+( a = 12 4) $%&!'( ζ = cos(2π
n)+ i sen(
2π
n) !+ #$" +()# ,6$ 7! )" ! #" ,6$ zn = 1/))"&"7" !"# %&' () % *0 8(7"+ )"+ +()# ,($!+ 7! )" ! #" ,6$ zn = 1 !+-9$ 7"7"+ *('*(-!$ ,"+ 7! ζ/ " +":!'/ *('
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3 + i 758 −2− 2i√3!708 sen(α) − i cos(α)! 7:8 1 + i tan(α) 7/8 1 + cos(α) + i sen(α)!;! "#$%&'(% (*( +,- *& .-' '/0+/&,1&' -3$.&4-' &, .( 5-%3( a + bi! <%- +%(% *(% %&'$+&'1('&#( 1('!7(8 5∠30◦ 798 2∠(7π/6) 7 8 4∠(−60◦)!7*8 √
2(cos(π/4) + i sen(π/4)!7&8 3(cos(π/6) + i sen(π/6))!
!! !"#$%&' ()* $+,-'.'/0$+#! !.!&#$, ! ! "#$$#% &$ '()*$+ , &$ #%-*'&./+ )& #)# *.+ )& $+1 12-*2&./&1 .3'&%+1 +'4$&5+1!6#7 3 + 4i! 687 5i! 6 7 −5i!6)7 1 +√3 i! 6&7 3− 5i! 697 −10 + 6i!:! ;<#$*#%6#7 (1− i)13 687 (1 + i
√3)5 6 7 (3 − 4i)−5!=! >&1+$<&% +'4$&/#'&./&6#7 z3 = 1 687 z3 = 8 6 7 z6 = 16)7 z4 = −1 6&7 z4 = −16 697 z8 = 1− i!?! @1#.)+ $+1 'A/+)+1 )&$ &5&'4$+ B !C!D 4%+8#% $#1 12-*2&./&1 2)&./2)#)&1E6#7 cos(3θ) = cos3(θ) − 3 cos(θ) sen2(θ)!687 sen(3θ) = 3 cos2(θ) sen(θ)− sen3(θ)!6 7 cos(4θ) = cos4(θ) − 6 cos2(θ) sen2(θ) + sen4(θ)!6)7 sen(4θ) = 4 cos( θ) sen(θ)(cos2(θ) − sen(θ))EF! 6>#G &1 382 #1 )& $# *.2)#)7 H&# ω $# %#GI 382 # 8J12 # )& $# *.2)#)E + 1&# ω = ∠(2π/3)!K%+8#% L*&6#7 K#%# #)# .3'&%+ .#/*%#$ mE 1& /2&.& L*& ωm /#'82A. &1 *.# %#GI 382 # )& $# *.2)#)!6H*-&%&. 2#M &<#$*#% (ωm)37!687 ωk+3 = ωE &1 )& 2% L*& $+1 <#$+%&1 )& ωm )29&%&./&1 1+. ωk +. k = 0, 1, 2!C! 6>#G &1 L*2./#1 )& $# *.2)#)7 H&# ζ = ∠(2π/5) , β = ζ + ζ−1N K%+8#% $+1 12-*2&./&1 &.*.O 2#)+1!6#7 ζ5 = 1!687 β2 + β − 1 = 0!6 7 β =
1
2(−1 +
√5)!6)7 ζ2 − βζ + 1 = 0!6&7 ζ =
1
2(β −
√
4− β2)!697 P+. $*2% L*&cos(72◦) =
−1 +√5
4sen(72◦) =
√
10 + 2√5
4.Q! 6>#G &1 nOA12'#1 )& $# *.2)#)7! H&# n *. .3'&%+ .#/*%#$ '#,+% L*& B! H&# ζ = ∠(2π/n)E +1&# $# %#GI 8J12 # nRA12'#!6#7 K%+8#% L*& 4#%# /+)+ .3'&%+ .#/*%#$ kE ζk = 1!
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cos(α) + i sen(α)5 !" cos(α) + i sen(α)
cos(α)− i sen(α)5 " 1
cos(2α) + i sen(2α)5 4" cos(4α)− i sen(4α)
cos(2α)− i sen(2α)5 )" (cos(
π
5) + i sen(
π
5))55 2" (cos(
3π
5) + i sen(
3π
5))55 <" (∠(5α))3(∠(α))−3
(∠(3α))2(∠(3α))2 E" 1− cos(2α) + i sen(2α)
1 + cos(2α)− i sen(2α)5>>5 6)& ζ (-& *%1( +8- !F*+ & 4) z125 =& 9$%9%*+ +8- >?5@5A /()*0$& '() 1&* 9%0)- +&* 4) ζ *%- 1&*$&G )* 12HI*+/&* 4) 1& (-+4&45 &" #$%!&$ '() 1&* 9%0)- +&* 4) ζ5 %-0+)-) & 1&* 4% ) $&G )* 12HI*+/&* 4) 1& (-+4&4J 9)$%'() 1% &-0)$+%$ -% )* +)$0% 9&$& ζ35 !" K)0)$/+-&$ 9&$& '() 7&1%$)* 4) mJ 1&* 9%0)- +&* 4) ζm %-0+)-)- & 0%4&* 1&* $&G )* 12HI*+/&* 4) 1& (-+4&45>A5 L&11&$ 0%4&* 1&* *%1( +%-)* 4) xn = aJ a > 0J , n = 2, 3, 4, 5, 65 !"#" $%&'( )*+ -.+/( 0/&-12- '(#$% &'( *+(,% *- '.$-,% (+ %+( .$% /(* $% &'( *+(,% 0,-*12* $%3 #$ 45'-$ 6736! 8',%9-$ .$%5-:4 $% 2, +9-$% 0+%*;.,% &'( *+(,% 0,-*12* $% <', 9$8;*=( 9*,(,( >$-*$2$% $0.* $ *+(,%3
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...........................................................................................................................................?*5'-$ 6736!@ ?'( *+(,% 0,-*12* $%A( 5,(,-$.B 2, *8+% <',C($ &'( *1( f ,% !"#$%# ' +( '( 0,-D+2+ *5'$. $ pB %%*B 0$-$ 9+2+ x %, '80., <',
f(x+ p) = f(x).
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••••
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x
y=H(x)
• • • • • • • •
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1
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csc(α) + cot(α)# '" tan2(θ) =
1
cos2(θ)− 1# (" 1
sec(α)− 1− 1
1 + sec(α)= 3 cot2(α)# )" sec2(β) − csc2(β) = tan2(β) − cot2(β)# *" sec2(γ) + csc2(γ) = sec2(γ) csc2(γ)# +" (cos(x)− sen(x))2 + (cos(x) + sen(x))2 = 2# ," 1 + cos(x)
sen(x)+
sen(x)
1 + cos(x)= 2 csc(x)# -" 1 + tan2(y)
tan2(y)= csc2(y)# ." 1− sen2(z)
sen2(z)= cot2(z)# /" sen4(α) − cos 4(α)
sen2(α) − cos2(α)= 1# 0" sen4(α) − cos 4(α)
sen(α) − cos(α)= sen(α) + cos(α)# 1" ln | sen(x)| − ln | cos(x)| = ln | tan(x)|# 2" 2 ln | cos(x)|+ 2 ln | sen(x)| = 0# 3" sec(u)
cot2(u)=
1− cos2(u)
cos2(u)# 4" 1 + sen(x) + cos(x)
1 + sen(x) − cos(x)=
1 + cos(x)
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