Self-assessment answers: 10 Trigonometric equations and

Mathematics Higher Level for the IB Diploma   

Copyright Cambridge University Press 2013. All rights reserved. Page 1 of 2  

Self-assessment answers: 10 Trigonometric equations and

identities

1. 3x = arcsin 1 π 5π

or 2 π

2 6 6 k

   

   for k ∈ .

 π 5π 13π 17π

3 , , , ,

6 6 6 6

x ...

 π 5π 13π 17π

, , ,

18 18 18 18

x [4 marks]

2. Using the identity sin² x = 1 – cos² x:

2sin² x – cos x = 1  2cos² x + cos x – 1 = 0

 (2cos x – 1)(cos x + 1) = 0

 cos x = 1 2 or –1

 x = π

3 , π [8 marks]

3. sin x (sin x − 3 cos x) = 0

 sin x = 0 or tan x = 3

 x = 0, 60, 240, 360 [6 marks]

4. (a)

1 1 cos2

cos cos cos

x x

x x

  

sin2

cos x

 x

sin tanx x



Mathematics Higher Level for the IB Diploma   

Copyright Cambridge University Press 2013. All rights reserved. Page 2 of 2  

(b) (i) 1 1

sin cos

sin cos

x x

x x

  

 1 1

sin cos

sin cos

x x

x x

  

 1 sin²

sin tan sin

x x x

x

  (by using your answer from (a))

 cos² x = sin² x tan x

 tan³ x = 1

 tan x = 1 (ii) tan x = 1  x = 5

4

 [12 marks]