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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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edmet-nklpunya.blogspot.comx

(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


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 =


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


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


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+=


(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

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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;

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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

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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]

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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]

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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]


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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]


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]

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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]


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]

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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]

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TAMAT