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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 =

 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

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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

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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 

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 

00

  −1

x

 

5

  2

  90 180 270 360

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(B)  The Basic Graph of Tangent

(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 

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 

0

−2

x

 

3. Complete the table below and sketch the graph of  y = tan 2xfor  0o

<  x  < 180o 

00

 

x

 

4. Complete the table below and sketch the graph of  y = 3 sin xfor  0o

<  x  < 360o 

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 

3

 

0

−3

x

 

7. Complete the table below and sketch the graph of  y = sin 2x − 1for  0o ≤  x  ≤ 180o 

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 

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]

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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