f5 c5 trigonometric functions new

7/30/2019 f5 c5 Trigonometric Functions New

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CHAPTER 5 TRIGONOMETRIC FUNCTIONS FORM 5

PAPER 1

106

1. Given is an acute angle andsin p = . Express each of the following in

terms ofp.

[3 marks]

a) tan

b) ( )cos 90oec

2. Given cos p = and 270 360o o .Express each of the following interms ofp.

[3 marks]

a) sec

b) ( )cot 90o

3. Given tan r = , where r is a constant and 180 270o o . Find interms ofr.

[3 marks]

a) cot

b) tan 2

4. Solve the equation 26cos 13cot 0ec x x = for 0 360o ox

[4 marks]5. Solve the equation 2 22sin cos sin 1A A A = + for 0 360o oA

[4 marks]

6. Solve the equation 2sin7cos2 2 = yy for 0 360o oy

[4 marks]

7. Solve the equation 2 015cos cos 4cos 60x x= + for 0 360o oA [4 marks]

8. Solve the equation 3cot 2sin 0x x+ = for 0 360o ox [4 marks]


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PAPER 2

107


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108

1. (a) Sketch the graph of sin2y x= for 0 180o ox [4 marks]

(b) (b) Hence, by drawing suitable straight line on the same axes, find the

number of solution to the equation1

sin cos2 360o

xx x = for

0 180o ox

[3 marks]

2. (a) Sketch the graph of 3cosy x= for 0 2x [4 marks]

(b) (b) Hence, by using the same axes, sketch a suitable graph to find the

number of solution to the equation2

3cos 0xx

+ = for 0 2x .

State number of solutions.

[3 marks]

3. (a) Sketch the graph of 3sin2y x= for 0 2x [4 marks]

(b) Hence, by using the same axes, sketch a suitable straight line to find

the number of solution to the equation 2 3sin 22

xx

= for

0 2x

[3 marks]

4.(a) Prove that

cot tancos 22

x xec x

+=

[2 marks]

(b) (i) Sketch the graph of3

2sin2

y x= for 0 2x [6 marks]

(ii) Find the equation of a suitable straight line to solve the equation

3 3 1sin

2 2 2x x

= .

Hence, on the same axes, sketch the straight line and state the

number of solutions to the equation3 3 1

sin2 2 2

x x

= for

0 2x .5. (a) Prove that 2 2 2sec 2cos tan cos 2x x x x = [2 marks]

(b) (i) Sketch the graph of cos2y x= for 0 2x [6 marks]

(ii) Hence, using the same axes, draw a suitable straight line to find

the number of solutions to the equation22cos 1

xx

= for

0 2x .

6. (a) Prove that 2 22 2sin 2cosx x = [2 marks]

(b) Sketch the graph of 12tan += xy for 20 x . By using the same

axes, draw the straight line9

32

y x

= and state the number of solution to

equation9

tan 2 22

x x

+ = for 20 x

[6 marks]

7. (a) Prove that 2 2 2 2cot cos tan secx ec x x x= + [2 marks]

(b) Sketch the graph3

cos2

y x= and 2siny x= for 0 2x . State

the number of solution to e uation1 3

sin cosx x = for

[6 marks]


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ANSWERS (PAPER 1)

109


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110

1 a)

2

tan1

p

p =

1

b)( )cos 90oec = ( )

1

sin 90o 1

= 21 p 1

2 a)sec =

1

cos

=1

p

1

b) ( )cot 90o = tan 1

=21 p

p

1

3 a)cot =

1

tan

=1

r

1

b)tan 2 =

2

2tan

1 tan

1

=2

2

1

r

r1

4 ( )26 1 cot 13cot 0x x+ = 1

26cot 13cot 6 0x x + =

( ) ( )3cot 2 2cot 3 0x x = 1

3cot 2 0x = OR 2cot 3 0x =

3tan

2x = OR

2tan

3x =

' 056 19 or 56.31

ox = and '33 41 or 33.69o ox = 1

' o ' ' o '56 19 , 236 19 33 41 , 213 41o ox = Or 56.31 , 236.31 ,33.69 ,213.69

o o o o

1

5 ( )2 22sin 1 sin sin 1A A A = + 1

2 22sin 1 sin sin 1A A A + = +

23sin sin 2 0A A =

( ) ( )3sin 2 sin 1 0A A+ = 1

( )3sin 2 0A + = OR ( )sin 1 0A =

2

sin3

A = OR sin 1A =

90 and 41.81o oA = 1

90 ,221.81 , 318.19o o o

A = 16 22cos 7sin 2 0y y + =

22(1 sin ) 7sin 2 0y y + = 1

22sin 7sin 4 0y y+ =


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(ANSWERS)PAPER 2

111


7/8

45o

135o

1

0.5

0.5

1

x

y

1180

o

xy =

180o

sin 2y x=


112

1 1 ( shape)

1(max/min)

1(one

period)

1(complete

from 0 to

180o)

1 (straight

line)

1sin cos

2 360ox

x x =

1

2 2 360oy x

=

1 180ox

y = 1Number of solutions= 3

1

2 a) 1 ( shape)

1(max/min)

1(one

period)

1(complete

from 0 to2 or

360 o)

1(for line

2y

x

=

b) 2 3cos 0xx

+ =

20y

x

=

(b)

2y

x

=

1

Number of solution =2

1

3 1 ( shape)

1(max/min)1(one

period)

1(complete

from 0o to

2 )

2

3

2

2

2

1

2

x

y

31y x

=

32sin

2y x=

y=2sinx2

2

3

2

2

2

1

1

2?

x

y

2

3

2

2

2

x

y

tan 2 1y x= +

2y

x

=

3cosy x=

-2


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113

f5 c5 trigonometric functions new

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