trigo module 9 - welcome to penang free school papers add maths/trigonometric... · additional...

TRIGONOMETRIC FUNCTION

ADDITIONAL MATHEMATICSFORM 5

MODULE 9

http://mathsmozac.blogspot.com

http://sahatmozac.blogspot.com

1

CONTENTS

CONTENTS PAGES

9.0 CONCEPT MAP 2

9.1 EXERCISE 1 3

9.2 PAST YEAR SPM QUESTION 5

9.3 ASSESMENT 6

ANSWER 7



2

9.0 CONCEPT MAP

TRIGONOMETRICFUNCTION

POSITIVE ANGLES AND NEGATIVE ANGLES

THE SIXTRIGONOMETRICFUNCTION

Special angles 30o,___ and 60o

Tan Ө = sin Ө

sec Ө = 1cos Ө

cosec Ө= 1sin Ө

cot Ө = cosec Ө

Basic Identities sin2 Ө + cos2 Ө 1

____+ tan2 Ө sec2 Ө

1 + cot2 Ө______

Complementary Angle

sin cos(90 )o

cos sin(90 )o

tan cot(90 )o

sec cos (90 )oec

cos sec(90 )oec



3

9.1 Exercise 1

1. Given that sin Ө = 8

17and cos Ө = -

15

17. Find the value of :-

a) sec Ө = 1

cos

=1

1517

= -17

15

b) cot Ө

2. Given that sin 36o = 0.5878, cos 36o = 0.8090 and tan36o = 0.7265 . Find the value of

a) cos 54 = sin (90 o – 54 o)= sin 36= 0.5878

b) cosec 54 o

3. Solve the trigonometric equation for 0 o x 360 o

a) cos 2x = - 0.7660

basic angle = 40 o

Given 0 o x 360 o

So 0 o 2x 720 o

Cos is negative at quarter II and III2x = 180 o - 40 o, 180 o + 40 o,

(180 o - 40 o ) + 360 o,(180 o + 40 o ) + 360 o,

= 140 , 220, 500, 580x = 70 o, 110 o, 250 o, 290 o

e) 2 sin x = tan x

b) tan x = cot 60 o f) 3 sin x cos x = 2 sin2 x



4

c) sin ( x – 30 o) = -1

2

g) 2 sin2 x + cos x – 2 = 0

d) sin x + 2 sin x cos x = 0sin x ( 1 + 2cos x) = 0sin x = 0

x = 0 o, 180 o, 360 o

or1 + 2cos x = 0

2cos x = -1

cos x = -1

2x = 120 o, 240 o

so x = 0 o, 120 o, 180 o, 240 o

h) tan2 x + sec x – 5 = 0

4. Prove that identity

a) 2 2 2 2cot tan cos secx x ec x x b)

21 2sincos sin

sin cos

xx x

x x



5

9.2 PAST YEAR SPM QUESTION

2003, Paper 1

1. Given that tan ,0 , 90o ot , express in terms of t :

) cot

)sin(90 )o

a

b

[3 marks]

2. Solve the equation 26sec 13tan 0,0 360o oA A A [ 4 marks]

2004, Paper 1

3. Solve the equation 2 2cos sin sinx x for 0 360o ox [ 4 marks]



6

9.3 ASSESSMENT

1. Given that3

cos5

x and 0 180o ox , find sec x + cosec x .

Solution:

2. Given that sin Ө = k and Ө is acute angle, express in term of k:a. tan Өb. cosec Ө

Solution:

3. Solve the equation 2 25sin 2 3 0,0 360o oA cos A A

Solution:

4. Prove that 2 2 2 2tan sin tan sinA A A A



7

ANSWER

Exercise 1

1. b) -15

82. b) 1.236

3. b) 30 ,210o ox c) 30 ,90 , 210 , 270o o o ox

e) 0 ,60 ,180 ,300 ,360o o o o ox f) 0 ,56 19 ,180 , 236 19 ,360o o o o ox

g) 60 ,90 , 270 ,300o o o ox h) 60 ,109 28 , 250 32 ,300o o o ox

Past Year SPM Question

1. a)1

tb)

2

1

1t

2. 33.69 ,56.31 , 213.69 ,236.31o o o oA

3. 30 ,150 , 270o o ox

Assesment

1.35

12

2. a)2

1

1 kb)

1

k

3. 35.26 ,144.74 ,180 ,215.26 ,324.74o o o o oA



8

TRIGONOMETRICFUNCTIONS

ADDITIONAL MATHEMATICSFORM 5

MODULE 10



9

Contents

10. 1 ADDITION FORMULAE AND DOUBLE-ANGLE FORMULAE

10.1.a Understand and use addition formulae and double-angle formulae

10.1.b Solving Trigonometric Equations by using Addition Formulae and

Double- angle Formulae

10.2 GRAPHS OF FUNCTIONS OF SINE, COSINE AND TANGENT

10.2.1 Sketch the graph of functions sine

10.2.2 Sketch the graph of functions cosine

10.2.3 Solution of equations involving the graphs

10.3 SPM QUESTIONS



10. TRIGONOMETRIC FUNCTIONS

10. 1 ADDITION FORMULAE AND DOUBLE-ANGLE FORMULAE

10.1.a Understand and use addition formulae and double-angle formulae

List down the Addition Formulae and Double-Angle FormulaeADDITION FORMULAE DOUBLE-ANGLE FORMULAE

Example

Given that sin A =5

4and cos B =

13

12where A and B are angles in the second

and fourth quadrants respectively. Calculate the value of tan ( B – A )

Solution

A in the second quadrant B in the fourth quadrant

( 12 , -5 )

-5

12B

13

( -3 , 4 )

45

A

- 3


sin A =5

4

tan A =3

4

http://mathsm

cos B =13

12

10

tan B =12

5

ozac.blogspot.com

11

tan ( B – A ) = BA

AB

tantan1

tantan

=

3

4

12

51

3

4

12

5

=

9

1412

11

=56

33

EXERCISE 1

1. Given that sin α =17

8, 90o < α < 270o and sin β =

13

12 , 90o < β < 270o.

Calculate the value of( a ) sin (α + β ) ( b ) cos ( β – α )

………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

2. Given that tan x =3

4, sin y =

5

4 where x and y are angles in the same quadrant.

Without using the calculator, calculate the value of( a ) sin ( x – y ) ( b ) cos ( x + y ) ( c ) tan ( y + 45o ) ( d ) tan ( x + y )

………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………



12

………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

3. Given that tan θ = p and θ is an acute angle. Express each of the following in terms of p

( a ) sin 2 θ ( b ) cos 2 θ ( c ) tan 2 θ

………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

10.1.b Solving Trigonometric Equations by using Addition Formulae and Double-angle Formulae

Example

Find all the angles between 0o and 360o which satisfy( a ) 2 sin 2 θ = sin θ( b ) 2cos2 θ + 3 = 8 sin θ



13

Solution

( a ) 2 sin 2 θ = sin θ 2 ( 2sin θ cos θ ) = sin θ 4sin θ cos θ - sin θ = 0 sin θ (4cos θ – 1 ) = 0 sin θ = 0 or 4cos θ-1=0

θ = 0 o , 180 o , 360 o cosθ = 4

1

θ = 75 o 31’ , 284o29’

So that , θ = 0 o , 75 o 31’ , 180 o , 284o29’, 360 o

( b ) 2cos2 θ + 3 = 8 sin θ 2 ( 1- 2sin2θ ) + 3 = 8 sin θ 2 - 4sin2θ + 3 = 8 sin θ 4sin2θ + 8 sin θ – 5 = 0 ( 2 sin θ – 1 )( 2 sin θ + 5 ) = 0 2 sin θ – 1 = 0 or 2 sin θ + 5 = 0

sin θ =2

1sin θ =

2

5

θ = 30 o , 150 o θ is undefinedSo that θ = 30 o , 150 o



14

EXERCISE 2

1. Find all the angles between 0o and 360o which satisfy( a ) 3sin 2A = 4sin A( b ) 5sin2A = 5 – sin 2A

………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

2. Find all the angles between 0o and 360o which satisfy

( a ) 3cos2α – 5 = 8 cos α( b ) tan 2α tan α = 1

………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………



15

10.2 GRAPHS OF FUNCTIONS OF SINE, COSINE AND TANGENT

10.2.1 Sketch the graph of each of the following functions

1 ) y = 2sin θ

2 ) y =2

1sin θ

3 ) y = - sin θ

4 ) y = sin 2θ5 ) y = sin

2

1θ

θO

y

180o

360o

270o

90o

y = sin θ

1

-1



17

10.2.2 Sketch the graph of each of the following functions

1 ) y = 2cos θ

2 ) y =2

1cos θ

3 ) y = - cos θ

4 ) y = cos 2θ5 ) y = cos

2

1θ

θO

y

180o

360o270

o

90o

y = cos θ

1

-1



19

10.2.3 SOLUTION OF EQUATIONS INVOLVING THE GRAPHS

Determining number of solutions by sketching the graphs

Example

Sketch the graph of the function y = cos 2θ for 0 < θ < 2π. Hence find the number ofsolutions of each of the equation

cos 2θ + 0.5 = 0

Solution

cos 2θ = - 0.5

The number of solutions = 4

y

θO π 2π

2

2

3

1

- 1

y = cos 2θ

y = - 0.5



20

Exercise 3

1. Sketch the graphs of the functions y = - 2sin 2x and y =2

xfor 0 < θ < 2π on the

same axes. Hence, state the number of solutions of the equation 4πsin 2x + x = 0.

2. Sketch the graphs of the functions y = 3cos2x and 2y + x = 0 for 0 < θ < 2π on the same axes. Hence, state the number of solutions of the equation 6 cos2x + x = 0.



21

10.3 SPM QUESTIONS

SPM 2003 PAPER 2 NO 8

( a ) Prove that tan θ + cot θ = 2 cosec 2θ [ 4 marks ]

( b ) ( i ) Sketch the graph y= 2 cos x2

3for 0 ≤ x ≤ 2π

( ii ) Find the equation of a suitable straight line for solving the equation

cos x2

3= 1

4

3x

Hence, using the same axes, sketch the straight line and state the number of

solutions to the equation cos x2

3= 1

4

3x

for 0 ≤ x ≤ 2π [ 6 marks ]



22


( a ) Sketch the graph of y = cos2x 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 [ 3 marks ]



23


( a ) Prove that cosec2x – 2 sin2x – cot2x = 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( cosec2x – 2sin2x – cot2x ) = 1

xfor 0 ≤ x ≤ 2π.

State the number of solutions [ 6 marks ]



24

10.4 ASSESSMENT

1. Given that sin θ = k such that θ is an acute angle, express cos 2θ in terms of k[ 2 marks ]

2. Find all values of x between 0o and 360o which satisfy the equation3 tan2y = cot y [ 3 marks ]

3. Sketch the graph of y = | sin 2x | for 0 ≤ x ≤ 2π. Determine the equation of a suitablestraight line to solve the equation x - 2π| sin 2x | = 0.Sketch the straight line and hence, state the number of solutions to the equationx - 2π| sin 2x | = 0 for 0 ≤ x ≤ 2π.

[ 5 marks ]



25

ANSWER:Exercise 1.

1. a)221

140b) -

221

21

2. a) 6 b ) -25

7d ) -

7

24

3. a)1

22 c

cb )

1

12

2

c

cc)

21

2

c

c

Exercise 2.

1) a) 0 0 , 48011’, 1800, 3110 480 , 3600

b ) 680 12’,900 , 248012’, 2700

2) a) 1310 49’ , 228011’b ) 300 , 1500 , 2100 , 3300

10.2

θO

y

180o

360o

270o

90o

y = sin2 θ

1

-1

θO

y

180o

360o

270o

90o

y = -sin θ

1

-1

θO

y

180o

360o

270o

90o

y =2

1sin θ

2

1

2

1

θO

y

180o

360o

270o

90o

y = 2sin θ

2

-2



26

O

y

180o

360o270

o90

o

y =| sin2 θ|

1

-1

θθO

y

180o

360o

270o

90o

y = 2sin2 θ

2

-2

θO

y

180o

360o

270o

90o

y = |sin θ|

1θ

y180

o360

o270o

90o

y = sin θ-1

-2

-1

y

180o

360o

270o

90o

y = sin θ+1

2

1θO

y

180o

360o

270o

90o

y = sin2

1θ

1

-1



27

θO

y

180o

360o270

o

90o

y = cos2θ

1

-1

θO

y

180o

360o270

o

90o

y =- cos θ

1

-1

θO

y

180o

360o270

o

90o

y =2

1cos θ

2

1

θO

y

180o

360o

270o

90o

y = 2cos θ

2

-2

y = |sin θ|+1

90o

180o

270o

360o

2



29

Exercise 31.X = 0 , 0.53 , 0.95 , 1.58 , 1.902.Number of intersection = 4

SPM 2003

Number of intersection = 3

1

y =| cos θ|+ 1

90o

180o

270o

360o

y

θO π 2π

1

- 1

y = - 0.5

y = 3cos x2

3

y = 22

3x

3

3

5



30

SPM 2004

(a ) (b )


SPM 2005


y

θO

4

2π

1

- 1

y = cos2x

3

5

θO

y

180o

90o

1

-1

y =180

x-1

y =3

1

3

x



10.4 ASSESSMENT1. 1 -2k2

2. y = 300, 1500,2100,3300

3.

,Number of intersection

O

y

90o

y = |2sin2 θ|

2


31

= 4

θ180

o

)(2 xy


trigo module 9 - welcome to penang free school papers add maths/trigonometric... · additional...

Documents