GCE O Level Past Exam Papers 1978-1987

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Additional Mathematics

Chapter -Trigonometry sin( A B)  si sin Aco cos B  cos Asi sin B

Addition Formulae

cos( A B)  co cos Aco cos B sin Asi sin B tan( A  B) 

cos 2 A cos

2

A sin

2

1 tan  A tan B

A

2

sin2 A 2sin Acos A

cos 2 A  1  2s 2 sin A

Double Angle Formulae

tan A  tan B

2

tan2 A 

cos 2 A  2 cos A  1 2

cos2 A 

1  tan  A

sin2 A 

2

1  tan  A

o

2tan  A

2tan  A

2

1  tan  A

o

1) Find all the angles between 0 and 360 which satisfy the questions 1

i)

cos

ii)

sin  y  2 cos y  0 .

3

 x

0.2

 

2) (a) Prove the identity 2

2

tan   sin 



2

2

tan  sin 

 

(b) If A, B and C are the angles of a triangle and tan  A  1 and tan B  2 show, without using the tables or calculators, tan C   3 .

3) Given that tan   

3 4

and that 180

   

270 , find, without using tables or

calculators, the value of  a)

tan tan 2 

b) tan3  c)

1 tan   2

For this particular value in (c) , verify that

1 2tan   2 sin    . 2 1 1  tan   2

2

1  tan  A

o

o

4) Find all the angles between 0 and 360 which satisfy the equations. a) cos 2 x  cos x  0 b) 4 tan 2 y tan y  1

5) Using the same axes and scales, sketch the curves  y  cos x and 0   x 

 

2

 y



cos2 x for

. Within this range of values of x, calculate the maximum distance between

the two curves, measured parallel to the y-axis. (Proof that the distance is a maximum is not required)

o

o

6) (a)Given that sin x  0.6 and 0
(b) Given that cos  y  0.72 and 0

  y  180

o

1 2

 x .

, find the value of  sin3 y .

o

7) Find all the angles between 0 and 360 inclusively which satisfy the equations, (a) 3sin x  tan x (b) 4 tan 2 y  1

8) Prove that (a) 2 sin(

A 45 

o

) cos(

A 45 

)

 cos 2

A

B 45 ) cos(  B 45 )  cos 2 (b) 2 cos( 

B

Hence solve the equation tan x tan 2 x  3 for 0

o

  x  360

o

9) (a) Find all the angles between 0 and 360 which satisfy the equation 3cos (c) Given that cos x cos y  0.25 and sin  x sin y 

5 8

 cot   .

, find the value of  cos( x  y )

and cos( x  y) .Hence, or otherwise, find the possible values of x and y between o

o

0 and 90 .