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16.1 POSITIVE & NEGATIVE ANGLES
(a) Positive angles – angles measured in the
anticlockwisedirection
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
2π
(f) − 4
7π
16.2 (A) THE SIX TRIGONOMETRIC FUNCTIONS(i) sin θ =
r
yif r = 1, then sin θ = y
(ii) cos θ =r
x if r = 1, then cos θ = x r y
(iii) tan θ = x
y=
θ
θ
cos
sin x
(iv) cosec θ = y
r if r = 1, then cosec θ = y
1 =θsin
1
(v) sec θ = x
r if r = 1, then sec θ =
x
1=
θcos
1
(vi) cot θ = y
x =
θtan
1=
θ
θ
sin
cos
16.2 (B) COMPLEMENT ANGLES
(i) sin θ = cos (90o− θ)
(ii) cos θ = sin (90o − θ)
(iii) tan θ = cot (90o − θ)
(iv) cosec θ = sec (90o − θ)
(v) sec θ = cosec (90o − θ)
θ
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(vi) cot θ = tan (90o − θ)
16.2 (C) RELATIONSHIPS BETWEEN ANGLES > 90O
AND ITS ACUTE ANGLES
Quadrant II: Quadrant IV:sin θ = sin (180o − θ) sin θ = − sin (360o − θ)
cos θ = − cos (180o − θ) cos θ = cos (360o − θ)
tan θ = − tan (180o − θ) tan θ = − tan (360o − θ)
Quadrant III: Note:sin (θ − 180o) = −sin θ If θ is the corresponding acute
cos (θ − 180o) = −cos θ angle in the quadrant, then angle
tan (θ − 180o) = tan θ in Quadrant III is (180 + θ).
16.2 (D) SPECIAL ANGLES: ( 0O, 30O, 45O, 60O, 90O, 180O, 270O, 360O)
θ
0O
30O
45O
60O
90O
180O
270O
360O
sin 02
11 0 − 1 0
cos 12
10 − 1 0 1
tan 0 1 ∞ 0 ∞ 0
Exercise 16.2:
1. Given that sin θ =5
3, find the value of each of the following
a) cos θ
2
QuadrantIII
Tangent
positive(θ −180o)
Quadrant IAll
positive
( θ )
QuadrantIV
Cosine
positive
(360o − θ)
QuadrantII
Sine
positive
(180o − θ)
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b) cosec θ
c) tan θ
d) cot θ
2. Given cosθ
=−
p and 180
o<
θ
<
270
o
, evaluate the following withoutusing a calculator.
(a) tan θ (b) sec θ
(c) sin θ (d) cosec θ
3. Given that sin 55o = 0.8192, cos 20o =
0.9397 , tan 55o = 1.4281 and cot 20o = 2.7473, find the following trigonometric expressions without
using a calculator.
(a) cot 35o (b) tan 70o
(c) sin 70o (d) cos 35o
4. Convert the following trigonometric expression to their corresponding
trigonometric expression in Quadrant I. Hence, evaluate their values.
(a) sin 120o (b) cos 200o
(b) tan (− 325o) (d) cot 350o
(d) cosec3
2π
(e) sec (− 4
)
5. Without using calculator, find the value of the following.
(a) sin 330o (b) cos 150o
3
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(c) tan (− 60o) (d) cot 225o
(e) sec (− 240o) (f) cosec 390o
6. Solve the following trigonometric equation for 0o
< θ < 360o
.
(a) sin θ = − 0.6428 (b) sec θ = 2
(c) cos2
1θ = 0.6690 (d) tan
2
1θ = − 0.25
(e) cosec 2θ = − 2.3662 (f) cot 2θ = sin 36o
(g) sin (θ + 30o) = 0.3566 (h) tan (2θ − 90o) = − 0.8300
7. Find all possible values of x for 0o< x < 360o without using calculator.
(a) tan x = cot 46o(b) cos x = sin (− 53o)
(c) sec x = cosec 35o 22’ (d) cosec x = − sec 82o 15’
8. Find all possible values of x for 0o< x < 360o without using calculator.
(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
4
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16.3 GRAPH OF SINE, COSINE AND TANGENT FUNCTIONS
(A) The Basic Graph of Sine
x (in radian) 02
ππ
2
3ππ2
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 x0 1 0 -1 0
y
1
00
−1
x
5
2
90 180 270 360
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(B) The Basic Graph of Tangent
x
(in degree)0o 45o 90o 135o 180o 225o 270o 315o 360o
y = tan x 0 1 ∞ −1 0 1 ∞ −1 0
y
00
x
1. Complete the table below and sketch the graph of y = sin 2xfor 0
< x < 2Π
y
1
0
−1
x
6
Exercise 16.3: 90 180 270 360
o
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2. Complete the table below and sketch the graph of y = 2 cos 2xfor 0o
< x < 360o
y
2
0
−2
x
3. Complete the table below and sketch the graph of y = tan 2xfor 0o
< x < 180o
y
00
x
4. Complete the table below and sketch the graph of y = 3 sin xfor 0o
< x < 360o
y
3
0 x
7
45o 90o 135o 180o
o
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−3
5. Complete the table below and sketch the graph of y = │2 sin x│for 0o ≤ x ≤ 360o
y
2
0
−2
x
6. Complete the table below and sketch the graph of y = │2 cos x│ + 1for 0o ≤ x ≤ 360o
y
3
0
−3
x
7. Complete the table below and sketch the graph of y = sin 2x − 1for 0o ≤ x ≤ 180o
y
2
1
0
−1
x
8
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1. Sketch the graphs of y = 2 cos x for 0 ≤ x ≤ π2 and y =π2
x on the
same axes. Hence determine the number of solutions for x between 0 and
π2 which satisfy the equation 2 cos x =π2
x .
x 0 π
y
y
2
0
−2
x
Number of solutions =
2. Sketch the graphs of y = │tan x│ for 0 ≤ x ≤ 2π and y = 1 −π3
2 x on
the same axes. Hence determine the number of solutions for x between
0 and π2 which satisfy the
equation │tan x│ = 1 −π3
2 x
x 0 π
9
Exercise 16.4 : Problem Solvin of Tri onometric Functions
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y
y
1
0
−1
x
Number of solutions =
3. Sketch the graphs of
y = 4 sin 2xfor
0
≤ x
≤π2 and
y = 1−
π2
3 x
on the same axes. Hence determine the number of solutions for x
between 0 and π2 which satisfy the equation 4 sin 2x = 1 −π2
3 x .
x 0 π
y
y
4
0
−4
x
10
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Number of solutions =
16.4 BASIC IDENTITIES
The 3 basic identities:
• sin2 x + cos2 x = 1• 1 + tan2 x = sec2 x• 1 + cot2 x = cosec2 x
16.5 ADDITION 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) = Btan Atan
Btan Atan
−+
1
• tan (A − B) = Btan Atan
Btan Atan
+−
1
• tan (A ± B) = Btan Atan
Btan Atan
1
±
16.6 DOUBLE 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 – 1
cos 2A = 1 – 2 sin2 A
• tan 2A = Atan
Atan
21
2
−11
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• Note :• Similarly, the formulae can be apply to create
HALF-ANGLE FORMULAE or other Addition Angle.
Exercise 16.4:
1. Prove the following identities;
(a) cot x + tan x = cosec x sec x (b) cos4 x – sin4 x = 1 – 2 sin2 x
(c)θ21
1
tan
= cos θ (d) θ
θ
θ
θ
θsec
sin
cos
cos
sin2
1
1 =+
++
(e) sec2θ + cosec2
θ = sec2 θ cosec2
θ
(e) 121
12
2
2
−+−
x cos
x tan
x tan
2. Solve the following trigonometric equations for 0 ≤ x ≤ 360o;
(a) 6 cos2 x − sin x − 5 = 0 (b) 3 sin2
x – 5 cos x – 1 = 0
(c) tan2 x – sec x = 1 (d) 3 cosec x + 9 = cot 2 x
(e) 3 sin x + 2 = cosec x (f) tan x + 1 = 2 cot x
Exercise 16.5:
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 (d)
oo
oo
tantan
tantan
54841
5484
+−
12
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(e) 2 cos2 22.5o – 1 (f) sin 75o
2. Given cos 2A =4
1and A is an acute angle. Determine the value of;
(a) cos 4A (b) cos A
(c) sin A (c) tan A
13
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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 (d) 3 cos 2x + cos x – 2 = 0
(e) 5 sin2 x = 5 – sin 2x (f) tan 2x = 4 cot x
(g) 1 (+ sin x)(3 + sin x) = 2 cos2 x (h) x sec2
4+ 3 cos x = cos 2x
PAST YEAR SPM QUESTIONS
PAPER 1 /2009:
16. Solve the equation 3sin x cos x – cos x = 0 for 0
o
≤ x ≤ 360
o
. [3 marks]
PAPER 1 /2008:
17. Given that sin θ = p, where p is a constant and 90o ≤ x ≤ 180o. Find in
terms of p:
(a) cosec θ,
(b) sin 2θ. [3 marks]
PAPER 1 /2007:
18. Solve the equation cot x + 2cos x = 0 for 0o ≤ x ≤ 360o.
14
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[4 marks]
PAPER 1 / 2006:
15. Solve the equation 15 sin2 x = sin x + 4 sin 30o for 0o ≤ x ≤ 360o.
[4 marks]
PAPER 1 / 2005:
17. Solve the equation 3 cos 2x = 8 sin x – 5 for 0o ≤ x ≤ 360o.
[4 marks]
PAPER 1 / 2004:
18. Solve the equation cos2 x – sin2 x = sin x for 0o ≤ x ≤ 360o.
[4 marks]
PAPER 1 / 2003:
20. Given that tan θ = t , 0 < θ < 90o, express, in terms of t;
(a) cot θ
(b) sin (90 − θ) [3 marks]
15
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PAPER 2 / 2003 / SECTION B:
8. (a) Prove that tan θ + cot θ = 2 cosec 2θ [4 marks]
(b) (i) Sketch the graph of y = 2 cos x 2
3for 0 ≤ x ≤ 2π.
(ii) Find the equation of a suitable straight line for solving the equation
cos 14
3
2
3 − x x π
. Hence, using the same axes, sketch the
straight line and state the number of solutions for the equation
cos 14
3
2
3 − x x π
for 0 ≤ x ≤ 2π. [6 marks]
PAPER 2 / 2004 / SECTION A:
3. (a) Sketch the graph of y = cos 2x for 0o ≤ x ≤ 180o. [3 marks]
(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
x
for 0o ≤ x ≤ 180o.
[3marks]
PAPER 2 / 2005 / SECTION A:
5. (a) Prove that cosec2 x – 2 sin2 x − cot 2 x = cos 2x . [2 marks]
(b) (i) Sketch the graph of y = cos 2x for 0 ≤ x ≤ 2π.(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 – cot 2 x) =π
x − 1 for 0 ≤ x ≤ 2π.
16
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State the number of solutions. [6 marks]
PAPER 2 / 2006 / SECTION A:
4. (a) Sketch the graph of y = − 2 cos 2x for 0 ≤ x ≤ 2π. [4 marks]
(b) Hence, using the same axis, sketch a suitable graph to find the number
of solutions to the equation x
π
+ 2 cos x = 0 for 0 ≤ x ≤ 2π.
State the number of solutions. [3 marks]
PAPER 2 / 2007 / SECTION A:
3. (a) Sketch the graph of y = |3cos 2x | for 0 ≤ x ≤ 2π. [4 marks]
(b) Hence, using the same axis, sketch a suitable graph to find the number
of solutions to the equation 2 - |3cos 2x | =π 2
x
for 0 ≤ x ≤ 2π.
State the number of solutions. [3 marks]
PAPER 2 / 2008 / SECTION A:
17
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4. (a) Prove that x
x
x
2tan2
sec2
tan2=
−
[2
marks]
(b) (i) Sketch the graph of y = − tan 2x for 0 ≤ x ≤ π.
(ii) Hence, using the same axis, sketch a suitable graph to find the
number of solutions to the equation 02
sec2
tan23=
−
+
x
x x
π
for
0 ≤ x ≤ π.
State the number of solutions. [6 marks]
PAPER 2 / 2009 / SECTION A:
4. (a) Sketch the graph of y =2
3cos 2x for 0 ≤ x ≤
2
3π. [3
marks]
(b) Hence, using the same axis, sketch a suitable straight line to find the
number of solutions to the equation2
32cos
3
4=− x x
π
for 0 ≤ x ≤
2
3
π.
State the number of solutions. [3 marks]
18