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My
AdditionalMathematicsModules
Form 5Topic: 16
(Version 2012)
REALISATIONby
NgKL(M.Ed.,B.Sc.Hons.Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH.)
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16.1 POSITIVE & NEGATIVE ANGLES
(a) Positive angles angles measured in the anticlockwise direction from the positive x-axis.(b) Negative angles angles measured in the clockwise direction from the positive x-axis.
Exercise 16.1Represent each of the following angles in a unit circle. Then, state(i) the quadrant in which the angles located,
(ii) the corresponding acute angle.
(a) 150o (b) 315o (c) 225o (d)
6
(e)3
2
(f) 4
7
16.2 (A) THE SIX TRIGONOMETRIC FUNCTIONS
(i) sin =r
yif r= 1, then sin =y
(ii) cos =r
xif r = 1, then cos =x r y
(iii) tan =x
y=
cos
sin x
(iv) cosec =y
rif r = 1, then cosec =
y
1= sin
1
(v) sec =x
rif r = 1, then sec =
x
1= cos
1
(vi) cot =y
x= tan
1=
sin
cos
16.2 (C) RELATIONSHIPS BETWEEN ANGLES > 90O
AND ITS ACUTE ANGLES
Quadrant II: Quadrant IV:sin = sin (180o) sin = sin (360o)
cos = cos (180o) cos = cos (360o)
2
S
CT
A
Quadrant III
Tangentpositive
(180o)
Quadrant IAll
positive
(
)Quadrant IV
Cosinepositive
(360o)
Quadrant IISine
positive
(180o
)
16.2 (B) COMPLEMENT ANGLES
(i) sin = cos (90o )
(ii) cos = sin (90o )
(iii) tan = cot (90o )
(iv) cosec = sec (90o )
(v) sec = cosec (90o )
(vi) cot = tan (90o )
90
90 -
Quadrant III:sin ( 180o) = sin
cos ( 180o) = cos tan
( 180o) = tan
180o
180o 360o
Note: If is thecorresponding acute angle inthe quadrant, then the actual
angle in Quadrant III is ( +180o).
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tan = tan (180o) tan = tan (360o)
16.2 (D) SPECIAL ANGLES: ( 0O, 30O, 45O, 60O, 90O, 180O, 270O, 360O)
0O 30O 45O 60O 90O 180O 270O 360O
sin 0 21
1 0
1 0
cos 12
1
0 1 0 1
tan 0 1 0 0
Exercise 16.2:
1. Given that sin =5
3, find the value of each of the
following
2. Given cos = p and 180o
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(d) tan ( 325o) (d) cosec3
2
(e) sec (4
)
5. Without using calculator, find the value of the following. (Use the concept of special angles)
(a) sin 330o (b) cos 150o (c) tan ( 60o)
(d) cot 225o (e) sec ( 240o) (f) cosec 390o
6. Solve the following trigonometric equation for 0o 360o.
(a) sin = 0.6428
(b) sec = 2 (c) cos2
1 = 0.6690 (f) cot 2 = sin 36
o
7. Find all possible values ofx for 0o x 360o.
(a) tanx = cot 46o (b) cosx = sin ( 53o) (c) secx = cosec 35o 22 (d) cosecx = sec 82o 15
8. Find all possible values ofxfor 0o x 360o.
4
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(a) cos x + 3 sin x cos x = 0 (b) 3 sin x = 4 sin2 x (c) 2 (sin x cos x ) = 5 cos x (d) 2 tan x = 7 cot x
16.3 GRAPH OF SINE, COSINE AND TANGENT FUNCTIONS
(A) The Basic Graph of Sine
x (in radian) 02
2
3 2
y = sin x 0 1 0 -1 0
y
1
00
1
x
(B) The Basic Graph of Cosine
x (in degree) 0o 90o 180o 270o 360o
y = cos x 0 1 0 -1 0
y
1
00
1
x
(C) The Basic Graph of Tangent
5
2
2
3 2
90 180 270 360
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(in degree)0o 45o 90o 135o 180o 225o 270o 315o 360o
y = tan x 0 1 1 0 1 1 0
y
1
0-1
x
1. Sketch the graph of y = sin 2x for 0 x 2 y
1
0
1
x
2. Sketch the graph of y = 2 cos 2x for 0o x 360oy
3. Sketch the graph of y = tan 2x for 0o x 180o
y
0 x
6
Exercise 16.3:
90 180 270 360
o
45o 90o 135o 180o
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4. Sketch the graph of y = 3 sin x for 0o x 360o
y
5. Sketch the graph of y = 2 sin x for 0o x 360oy
6. Sketch the graph of y = 2 cos x + 1 for 0o x 360oy
7. Sketch the graph of y = sin 2x 1 for 0o x 180oy
7
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8. Sketch the graph of y = cos x + 2 for 0o x 360oy
1. Sketch the graphs of y = 2 cos x for 0 x 2 and y =2x
on the same axes. Hence determine the number of
solutions forx between 0 and 2 which satisfy the equation 2 cos x = 2x
.
y
2. Sketch the graphs of y = tan x for 0 x 2 and y = 1 32x
on the same axes. Hence determine the number of
solutions forx between 0 and 2 which satisfy theequation tan x = 1 3
2x
y
8
Exercise 16.4 : Problem Solving involving the Six Trigonometric Functions
Number of solutions =
Number of solutions =
x 0 y
x 0 y
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3. Sketch the graphs of y = 4 sin 2x for 0 x 2 and y = 1 23x
on the same axes. Hence determine the number of
solutions forx between 0 and 2 which satisfy the equation 4 sin 2x = 1 23x
.
y
x
16.4BASIC IDENTITIES
The 3 basic identities: sin2 x + cos2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = cosec2 x
16.5ADDITION FORMULAE
sin (A + B) = sin A cos B + cos A sin B sin (A B) = sin A cos B cos A sin B sin (A B) = sin A cos B cos A sin B
cos (A + B) = cos A cos B sin A sin B cos (A B) = cos A cos B + sin A sin B cos (A B) = cos A cos B sin A sin B
tan (A + B) =BtanAtan
BtanAtan
+
1
tan (A B) =BtanAtan
BtanAtan
+
1
tan (A B) =BtanAtan
BtanAtan
1
Exercise 16.5: (Basic Identities)
1. Prove the following identities;
(a) cot x + tan x = cosec x sec x (b) cos4 x sin4x = 1 2 sin2x
9
x 0 y
Number of solutions =
16.6DOUBLE ANGLE FORMULAE
sin 2A = 2 sin A cos A
cos 2A = cos2A sin2A
Applying identity cos2 A + sin2 A = 1,
then, cos 2A = 2 cos2
A 1cos 2A = 1 2 sin2 A
tan 2A =Atan
Atan
21
2
Note: Similarly, the formulae can be apply to create
HALF-ANGLE FORMULAE or otherAddition Angle.
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xcosxsinxsinxcos
x2
sin21+=
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(c) 211
tan= cos (d)
secsin
cos
cos
sin2
1
1 =+++
(e) sec2 + cosec2 = sec2 cosec2 (f) 121
12
2
2
+
xcos
xtan
xtan
(g) (1 + cos )( 1 sec ) = sin tan (h)
2. Solve the following trigonometric equations for 0 x 360o. (Using the concept of basic identities)
(a) 6 cos2 x sin x 5 = 0 (b) 3 sin2x 5 cos x 1 = 0
10
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(c) tan2x sec x = 1 (d) 3 cosec x + 9 = cot2 x
(e) 3 sin x + 2 = cosec x (f) tan x + 1 = 2 cot x
Exercise 16.5: ( Use The Concepts of Double Angles )
1. Without using a calculator, find the value for the following trigonometric expression.
(a) sin 21o cos 24o + cos 21o sin 24o (b) tan 15o (c) cos 200o cos 65o + sin 200o sin 65o
(e) 2 cos2 22.5o 1 (f) sin 75o (f)oo
oo
tantan
tantan
54841
5484
+
2. Given cos 2A =4
1and A is an acute angle. Determine the value of;
11
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(a) cos 4A (b) cos A (c) sin A (d) tan A
3. Find all the values of x which satisfy the following trigonometric equations for 0o x 360o
(a) cos 2x 3 sin x + 1 = 0 (b) 3 tan x = 2 sin 2x
(c) cos 2x + cos2 x = 2 cos x (c) 3 cos 2x + cos x 2 = 0
(e) 5 sin2x = 5 sin 2x (f) tan 2x = 4 cot x
12
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(g) 1 (+ sin x)(3 + sin x) = 2 cos2 x(h)
xsec2
4+ 3 cos x = cos 2x
PAPER 1 /2009:
1. Solve the equation 3sin x cos x cos x = 0 for 0o x 360o.[3 marks]
PAPER 1 /2008:
2. Given that sin = p, wherep is a constant and 90o x 180o. Find in terms ofp:
(a) cosec ,
(b) sin 2. [3
marks]
13
PAST YEAR SPM QUESTIONS
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PAPER 1 /2007:
3. Solve the equation cot x + 2cos x = 0 for 0o x 360o.[4 marks]
PAPER 1 / 2006:
4. Solve the equation 15 sin2 x = sin x + 4 sin 30o for 0o x 360o.[4 marks]
PAPER 1 / 2005:
5. Solve the equation 3cos 2x = 8 sin x 5 for 0o x 360o.
[4 marks]
PAPER 1 / 2004:
6. Solve the equation cos2 x sin2x = sin x for 0o x 360o.
[4 marks]
14
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PAPER 1 / 2003:
7. Given that tan = t, 0
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(b) Hence, by drawing a suitable straight line on the same axes, find the number of solutions satisfying the equation
2 sin2 x = 2 180
xfor 0o x 180o.
[3marks]
PAPER 2 / 2005 / SECTION A:
10. (a) Prove that cosec2 x 2 sin2 x cot2 x = cos 2x. [2marks]
(b) (i) Sketch the graph ofy = cos 2x for 0 x 2.(ii) Hence, using the same axes, draw a suitable straight line to find the number of solutions to the equation
3(cosec2 x 2 sin2 x cot2 x) =x
1 for 0 x 2. State the number of solutions.[6 marks]
PAPER 2 / 2006 / SECTION A:
16
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11. (a) Sketch the graph ofy = 2 cos 2x for 0 x 2. [4marks]
(b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the equationx
+ 2 cos x = 0 for
0 x 2. State the number of solutions. [3marks]
PAPER 2 / 2007 / SECTION A:
12. (a) Sketch the graph ofy = |3cos 2x | for 0 x 2. [4marks]
(b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the equation 2 - |3cos 2x | =2
x
for 0 x 2. State the number of solutions.[3 marks]
17
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PAPER 2 / 2008 / SECTION A:
13. (a) Prove that xx
x2tan
2sec2
tan2=
[2
marks]
(b) (i) Sketch the graph of y = tan 2x for 0 x .(ii) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the equation
0
x2
sec2
2tanx
3x=
+ for 0 x . State the number of solutions. [6marks]
PAPER 2 / 2009 / SECTION A:
14. (a) Sketch the graph ofy =2
3
cos 2x for0 x 2
3 . [3
marks](b) Hence, using the same axis, sketch a suitable straight line to find the number of solutions to the equation
2
3cos2xx
3
4= for 0 x
2
3 . State the number of solutions. [3marks]
18
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PAPER 2 / 2011 / SECTION A:
15. (a) Sketch the graph ofy = 3 sin2
3
x for0 x 2. [4
marks](b) Hence, using the same axis, sketch a suitable straight line to find the number of solutions to the equation
0xsin3 =+2
3
x
for 0 x 2. State the number of solutions. [3marks]
19
TAMAT
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