# 7.04 Trigonometric equations

Worksheet
Trigonometric equations
1

Find the size of the acute angle \theta in the following equations:

a

\cos \theta = \dfrac{1}{2}

b

\sin \theta = \dfrac{1}{\sqrt{2}}

c

\tan \theta = \sqrt{3}

d

\sin \theta = \dfrac{\sqrt{3}}{2}

e

\sin \theta = \dfrac{1}{2}

f

\tan \theta = \dfrac{1}{\sqrt{3}}

2

Solve the following equations for 0\degree \leq x \leq 90\degree:

a

\tan x = 1

b

\sin x = \dfrac{1}{2}

c

2 \cos x = 1

d

\cos x = \dfrac{\sqrt{3}}{2}

e

\cos x - 1 = 0

f

\sqrt{2} \sin x = 1

g

\sin x - 1 = 0

h

\sin x = \dfrac{\sqrt{3}}{2}

3

Solve the following equations for 0\degree \leq \theta \leq 90\degree:

a

2 \sin \theta = 2 \sqrt{3} \cos \theta

b

\tan \theta = 2 \sqrt{3} - \tan \theta

c

6 \sin \theta - 3 \sqrt{3} = 0

d
4 \cos \theta - 2 \sqrt{3} = 0
4

State whether the following equations have a solution:

a

\cos \theta - 4 = 0

b

9 \tan \theta + 4 = 0

5

State the number of solutions for \theta in the following equations:

a

\cos \theta = - \dfrac{1}{2} for 0 \degree < \theta < 180 \degree

b

\tan \theta = - 1 for 0 \degree < \theta < 90 \degree

c

\sin \theta = - \dfrac{1}{\sqrt{2}} for 0 \degree < \theta < 90 \degree

d

\sin \theta = \dfrac{\sqrt{3}}{2} for 0 \degree < \theta < 180 \degree

6

Solve the following equations for 0\degree \leq x \leq 360\degree:

a

\cos x = \dfrac{1}{2}

b

\sin x = \dfrac{1}{\sqrt{2}}

c

\tan x = \dfrac{1}{\sqrt{3}}

d

\sin x = - \dfrac{1}{\sqrt{2}}

7

Solve the following equations for 0 \degree \leq \theta \leq 360 \degree:

a

\cos \theta = - \dfrac{1}{\sqrt{2}}

b

\cos \theta = 0

c

\sin \theta = 0

d

\cos \theta = -\dfrac{1}{\sqrt{2}}

e

\sin \theta = - \dfrac{\sqrt{3}}{2}

f

\sin \theta = 1

g

\tan \theta = \sqrt{3}

h

\tan \theta = 0

i

\tan \theta = - \dfrac{1}{\sqrt{3}}

8

Solve the following equations for 0 \degree \leq \theta \leq 360 \degree:

a

4 \tan \theta + 2 = - 2

b

8 \cos \theta - 4 = 0

c

2 \cos \theta + 4 = 3

d

8 \sin \theta - 4 \sqrt{2} = 0

9

Find the value of y in the following equations:

a

\dfrac{5}{8} \cos y = \dfrac{5 \sqrt{2}}{16} for 0 \degree < y < 90 \degree

b

\dfrac{6}{2} \sin y = \dfrac{3 \sqrt{3}}{2} for 0 \degree < y < 180 \degree

Non-exact value equations
10
Solve the following equations for 0 \degree \leq \theta \leq 90 \degree:
a

\cos \theta = 0.7986

b

\sin \theta =0.6428

c

\tan \theta =0.7265

d

\sin \theta = 0.3584

e

\tan \theta = 2.2460

11

Solve the following equations for 0 \degree \leq \theta \leq 360 \degree:

a

\cos \theta = 0.9063

b

\cos \theta = - 0.7986

c

\sin \theta = - 0.6428

d

\sin \theta = 0.9336

e

\tan \theta = 0.7002

f

\tan \theta = - 0.7265