Download Self-assessment answers: 9 Trigonometric equations and identities

Mathematics Standard Level for the IB Diploma Self-assessment answers: 9 Trigonometric equations and
identities
 1  π 5π
 2kπ for k ∈ .
1. 3x = arcsin    or
6
2 6
 3x 
 x
π 5π 13π 17π
, ,
,
, ...
6 6 6
6
π 5π 13π 17π
, ,
,
18 18 18 18
[4 marks]
2
2
2. Using the identity sin x = 1 – cos x:
2
2
2sin x – cos x = 1  2cos x + cos x – 1 = 0
 (2cos x – 1)(cos x + 1) = 0
 cos x =
x=
1
or –1
2
π
, π
3
3. sin x (sin x −
[8 marks]
3 cos x) = 0
 sin x = 0 or tan x =
3
 x = 0, 60, 240, 360
[6 marks]
2
4. Using double angle formula, cos 2θ = 1 – 2 sin θ
2
 2 sin θ + sin θ – 1 = 0
 (2 sin θ + 1)(sin θ – 1) = 0
 sin θ = 
1
or 1
2
 θ = 90, 210 or 330
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[6 marks]
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