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My

AdditionalMathematicsModules

Form 5Topic:16(Version 2010)

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 theanticlockwise

direction

from the positive x-axis.

(b) Negative angles angles measured in the clockwise directionfrom 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 = xy

=

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

2

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(vi)cot =y

x= tan

1=

sin

cos

16.2 (B) COMPLEMENT ANGLES

(i) sin

= cos (90

o

)(ii) cos = sin (90o)

(iii)tan = cot (90o)

(iv)cosec = sec (90o)

(v) sec = cosec (90o)

(vi)cot = tan (90o)

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)

tan = tan (180o) tan = tan (360o)

Quadrant III: Note:sin (180o) = sin If is the corresponding acute

cos (180o) = cos angle in the quadrant, then angle

tan (180o) = tan in Quadrant III is (180 + ).16.2 (D) SPECIAL ANGLES: ( 0O, 30O, 45O, 60O, 90O, 180O, 270O, 360O)

0O 30O 45O 60O 90O 180O 270O 360O

sin 02

11 0 1 0

3

90o

S

CT

A

Quadra

ntIII

Tange

nt

positive

(1

80o)

Quadran

tI

All

positive

(

)

Quadra

ntIV

Cosine

p

ositive

(360

o)

Quadra

ntII

Sine

positive

(180o

)

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cos 12

10 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

(a) cos (b) cosec

(c) tan (d) cot

2. Given cos = p and 180o

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(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

2(e) sec (

4

)

5. Without using calculator, find the value of the following.

(a) sin 330o (b) cos 150o

(c) tan (60o) (d) cot 225o

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(e) sec (240o) (f) cosec 390o

6. Solve the following trigonometric equation for 0o

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(g) sin ( + 30o) = 0.3566 (h) tan (290o) = 0.8300

7. Find all possible values ofx for 0o< x< 360o without using calculator.

(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 without using calculator.

(a) cos x + 3 sin x cos x = 0 (b) 3 sin x = 4 sin2 x

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(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

8

2

2

3 2

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1

00

1

x

(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< 2

y

1

9

Exercise 16.3:

90 180 270 360

90 180 270 360

o

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0

1

x

2. Complete the table below and sketch the graph of y = 2 cos 2x

for 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

0

0

x

4. Complete the table below and sketch the graph of y = 3 sin x

10

45o 90o 135o 180o

o

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

y

2

1

0

1

2

x

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 =2

x.

x 0 y

y

2

0 x12

Exercise 16.4 : Problem Solvin of Tri onometric Functions

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2

Number of solutions =

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 and2 which satisfy the

equation tan x = 1 32x

x 0

y

y

1

0 x

13

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1

Number of solutions =

3. Sketch the graphs of y = 4 sin 2x for 0 x 2 and y = 1 2

3x on

the same axes. Hence determine the number of solutions forx between 0

and 2 which satisfy the equation 4 sin 2x = 1 23x

.

x 0 y

y

4

0 x14

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4

Number of solutions =

16.4 BASIC IDENTITIES

The 3 basic identities:

sin2 x + cos2 x = 11 + tan2 x = sec2 x1 + cot2 x = cosec2 x

16.5 ADDITION FORMULAEsin (A + B) = sin A cos B + cos A sin B

sin (A B) = sin A cos B cos A sin Bsin (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

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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

Atan21

2

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 sin4x = 1 2 sin2x

(c) 211

tan= cos (d)

secsin

cos

cos

sin2

1

1 =++

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(e) sec2 + cosec2 = sec2 cosec2

(e) 121

12

2

2

+

xcosxtan

xtan

2. Solve the following trigonometric equations for 0 x 360o;

(a) 6 cos2 x sin x 5 = 0 (b) 3 sin2x 5 cos x 1 = 0

(c) tan2x sec x = 1 (d) 3 cosec x + 9 = cot2 x

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(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) oooo

tantan

tantan

54841

5484

+

(e) 2 cos2 22.5o 1 sin 75o

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2. Given cos 2A = 4

1

and A is an acute angle. Determine the value of;

(a) cos 4A (b) cos A

(c) sin A 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 3 tan x = 2 sin 2x

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(c) cos 2x + cos2 x = 2 cos x 3 cos 2x + cos x 2 = 0

(e) 5 sin2x = 5 sin 2x (f) tan 2x = 4 cot x

(g) 1 (+ sin x)(3 + sin x) = 2 cos2 x(h)

xsec2

4+ 3 cos x = cos 2x

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PAST YEAR SPM QUESTIONS

PAPER 1 /2009:

16. Solve the equation 3sin x cos x cos x = 0 for 0o x 360o.[3 marks]

PAPER 1 /2008:

17. Given that sin = p, wherep is a constant and 90o x 180o. Findin terms ofp:

(a) cosec ,

(b) sin 2. [3 marks]

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PAPER 1 /2007:

18. Solve the equation cot x + 2cos x = 0 for 0o x 360o.[4 marks]

PAPER 1 / 2006:

Solve the equation 15 sin2 x = sin x + 4 sin 30o for 0o x 360o.

[4 marks]

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PAPER 1 / 2005:

17. Solve the equation 3cos 2x = 8 sin x 5 for 0o x 360o.

[4 marks]

PAPER 1 / 2004:

18. Solve the equation cos2 x sin2x = sin x for 0o x 360o.

[4 marks]

PAPER 1 / 2003:

20. Given that tan = t, 0 < < 90o, express, in terms of t;(a) cot

(b) sin (90 ) [3 marks]

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PAPER 2 / 2004 / SECTION A:

3. (a) Sketch the graph ofy = 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

xfor

0o x 180o.

[3marks]

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PAPER 2 / 2005 / SECTION A:

5. (a) Prove that cosec2 x 2 sin2 x cot2 x = cos 2x. [2 marks]

(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]

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PAPER 2 / 2006 / SECTION A:

4. (a) Sketch the graph ofy = 2 cos 2x for 0 x 2. [4 marks]

(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. [3 marks]

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PAPER 2 / 2008 / SECTION A:

4. (a) Prove that xx

x

2tan2

sec2

tan2=

[2 marks]

(b) (i) Sketch the graph ofy = 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

2sec2

tan23=

+

x

xx

for

0 x .

State the number of solutions. [6 marks]

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PAPER 2 / 2009 / SECTION A:

4. (a) Sketch the graph ofy =2

3cos 2x for 0 x

2

3. [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= xx

for 0 x 2

3.

State the number of solutions. [3 marks]