A-Level Practise Card
Trigonometry
Q1. State whether the following trigonometric identities are true or false:
sin x
cot x
a) cot x =
b)
= cos x
cos x
cosec x
c) cosec2x = 1 + cot2 x
d) (sec x − tan x)(sec x + tan x) = 1
tan2 x − 1
e) 1 + 2 cos x =
tan2 x + 1
f) tan(3x) + tan2 x tan(3x) = sec2 x
2
Q2. Simplify the following expressions as much as possible:
1 − cos2 x
a)
sin2 x
d) cos x + sin x tan x
sin x cos y + sin y cos x
cos x cos y − sin x sin y
1 − tan2 x
e)
−1
1 + tan2 x
c) tan x cos x sin x
b)
f)
cos(2x)
2
+
cos2 x
cot2 x
Q3. Give all solutions in the range 0 ≤ x ≤ 2π for the following:
√
a) 2 cos x = 3
d) 2 sin(3x) = 1
√
b) 3 tan x = 3
√
e) 3 cot(2x) = −3
c) sin x + 2 = 3
f) tan2 x − 3 = 0
Q4. Give all solutions in the range 0 ≤ x ≤ 2π for the following:
3
a) 2 cos(3x) −
= −5
cos(3x)
4
c)
+ 3 cos x = 2 cot x tan x
sec2 x
b)
2
1 − 2 cos x sin(2x) = 0
d) cos(2x) = 1 − 2 sin2 x
Trigonometry (Answers)
A1. The answers are as follows:
a) false
b) true
c) true
d) true
e) false
f) false
A2. The simplified versions are as follows:
a) 1
b) tan(x + y)
c) sin2 x
d) sec x
e) cos(2x) − 1
f) sec2 x
A3. The solutions in the given range are as follows:
a)
π 11π
6, 6
b)
π 7π
6, 6
c)
π
2
d)
π 5π 13π 17π 25π 29π
18 , 18 , 18 , 18 , 18 , 18
e)
5π 11π 17π 23π
12 , 12 , 12 , 12
f)
π 2π 4π 5π
3, 3 , 3 , 3
A4. The solutions in the given range are as follows and are approximated to
six decimal places where appropriate:
a)
π 5π 7π 11π 13π 17π
9, 9 , 9 , 9 , 9 , 9
5π 3π 7π
b) 0, π4 , π2 , 3π
4 , π, 4 , 2 , 4 , 2π
c) 1.131403, 5.151782
d) this identity is true, so every value of x is a solution
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