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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+=

    edmet-nklpunya.blogspot.com

    (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