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5.06 Solving trigonometric equations 2

Worksheet
Compound angle equations
1

Consider the equation \cos \left(\theta + 20 \degree\right) = \dfrac{1}{\sqrt{2}} for 0 \degree \leq \theta \leq 360 \degree.

a

If \alpha = \theta + 20 \degree, find the solutions of \cos \alpha = \dfrac{1}{\sqrt{2}} for 20 \degree \leq \alpha \leq 380 \degree.

b

Hence, solve \cos \left(\theta + 20 \degree\right) = \dfrac{1}{\sqrt{2}}.

2

Consider the equation \cos \left(\theta - 100 \degree\right) = \dfrac{1}{2} for 0 \degree \leq \theta \leq 360 \degree.

a

If \alpha = \theta - 100 \degree, find the solutions of \cos \alpha = \dfrac{1}{2} for - 100 \degree \leq \alpha \leq 260 \degree.

b

Hence, solve \cos \left(\theta - 100 \degree\right) = \dfrac{1}{2}.

3

Consider the equation \sin \left(\theta + 35 \degree\right) = \dfrac{1}{2} for 0 \degree \leq \theta \leq 360 \degree.

a

If \alpha = \theta + 35 \degree, find the solutions of \sin \alpha = \dfrac{1}{2} for 35 \degree \leq \alpha \leq 395 \degree.

b

Hence, solve \sin \left(\theta + 35 \degree\right) = \dfrac{1}{2}.

4

Consider the equation \sin \left(\theta + 25 \degree\right) = \dfrac{\sqrt{3}}{2} for 0 \degree \leq \theta \leq 360 \degree.

a

If \alpha = \theta + 25 \degree, find the solutions of \sin \alpha = \dfrac{\sqrt{3}}{2} for 25 \degree \leq \alpha \leq 385 \degree.

b

Hence, solve \sin \left(\theta + 25 \degree\right) = \dfrac{\sqrt{3}}{2}.

5

Consider the equation \tan \left(\theta + 15 \degree\right) = \dfrac{1}{\sqrt{3}} for 0 \degree \leq \theta \leq 360 \degree.

a

If \alpha = \theta + 15 \degree, find the solutions of \tan \alpha = \dfrac{1}{\sqrt{3}} for 15 \degree \leq \alpha \leq 375 \degree.

b

Hence, solve \tan \left(\theta + 15 \degree\right) = \dfrac{1}{\sqrt{3}}.

6

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

a

\cos \left(\theta + 60 \degree\right) = 1

b

\cos \left(\theta - 75 \degree\right) = - \dfrac{1}{\sqrt{2}}

c

\sin \left(\theta + 45 \degree\right) = - \dfrac{\sqrt{3}}{2}

d

\tan \left(\theta - 45 \degree\right) = \dfrac{1}{\sqrt{3}}

e

\cos \left(\theta + 30 \degree\right) = -\dfrac{1}{2}

f

2\sin \left(\theta - 60 \degree\right) = -\sqrt{3}

7

Solve the following equations for 0 \degree \leq \theta \leq 360 \degree. Round your answers to two decimal places if necessary.

a

\cos 3 \theta = \dfrac{1}{\sqrt{2}}

b

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

c

\cos 2 \theta = 0.7

d

2 \sin 3 \theta - \sqrt{2} = 0

e

\sin \left(\dfrac{\theta}{2}\right) = \dfrac{\sqrt{3}}{2}

f

\tan 3 \theta = - 1

8

Solve the following equations for 0 \degree \leq x \leq 180 \degree:

a

\tan 4 x = \sqrt{3}

b

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

c

\cos 3 x = - \dfrac{1}{\sqrt{2}}

d

\tan\left(\dfrac{x}{3}\right) = 1

9

Solve the following equations for -180 \degree \leq x \leq 180 \degree:

a

\cos 3 x = - \dfrac{1}{\sqrt{2}}

b

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

c

\tan\left(\dfrac{x}{2}\right) = \pm \sqrt{3}

d

\sin 3x - 1 = 0

Squared equations
10

Solve the following equations for 0 \degree \leq \theta \leq 360 \degree. Round your answers to two decimal places if necessary.

a

2 \sin ^{2}\theta = 1

b
\sin^2 \theta = \dfrac{3}{4}
c
\cos^2 \theta = \dfrac{1}{2}
d
\tan^2 \theta = \dfrac{1}{3}
e
4 \sin^2 \theta = 1
f

\cos^{2}\left(\dfrac{\theta}{2}\right) - 1 = 0

g
\sin^2 2\theta = \dfrac{1}{2}
h

\cos^{2} 3\theta = \dfrac{3}{4}

11

Solve the following equations for the given domain:

a

\cos ^{2}\theta = \dfrac{3}{4} for 0 \degree \lt \theta \lt 90 \degree

b

4 \sin ^{2}\theta = 3 for 90 \degree \leq \theta \leq 270 \degree

Factorisation
12

State the number of solutions for \theta of the equation \left(\sin \theta + \dfrac{\sqrt{3}}{2}\right) \left(\cos \theta + \dfrac{\sqrt{3}}{2}\right) = 0 for 0 \degree \lt \theta \lt 90 \degree.

13

Solve the following equations for 0 \degree \leq \theta \leq 360 \degree. Round your answer to one decimal place if necessary.

a

\tan ^{2}\left(\theta\right) - 2 \tan \theta + 1 = 0

b

4 \cos ^{3}\left(\theta\right) = 3 \cos \theta

c

5 \sin ^{2}\left(\theta\right) + 8 \sin \theta \cos \theta - 4 \cos ^{2}\left(\theta\right) = 0

d

7 \cos ^{2}\left(\theta\right) + 3 \cos \theta = 3

e

9 \sin ^{2}\left(x\right) + 5 \sin x = 3

f

\cos \theta \tan \theta = \cos \theta

g

8 \tan ^{2}\left(\theta\right) \cos \theta - 4 \tan ^{2}\left(\theta\right) = 0

14

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

a

\left(\sin \theta - \dfrac{\sqrt{3}}{2}\right) \left(\cos \theta - \dfrac{1}{\sqrt{2}}\right) = 0

b

\left(\sin \theta + \dfrac{1}{\sqrt{2}}\right) \left(\tan \theta - \dfrac{1}{\sqrt{3}}\right) = 0

15

Solve the following equations for the given domain:

a

\left(\tan \theta - \dfrac{1}{\sqrt{3}}\right) \left(\cos \theta + \dfrac{1}{\sqrt{2}}\right) = 0 for 180 \degree \leq \theta \leq 270 \degree

b

\left(\sin \theta - \dfrac{1}{2}\right) \left(\cos \theta - \dfrac{1}{\sqrt{2}}\right) = 0 for 270 \degree \leq \theta \leq 360 \degree

c

\tan ^{2}\left(\theta\right) = \sqrt{3} \tan \theta for - 90 \degree \lt \theta \lt 90 \degree

d

2 \sin ^{2}\left(\theta\right) - \sin \theta = 0 for - 180 \degree \leq \theta \leq 180 \degree

16

Deborah is solving the equation 2 \sin^{2} \theta + 7 \sin \theta + 5 = 0. After some factorisation, she arrives at the pair of equations \sin \theta + 1 = 0 and 2 \sin \theta + 5 = 0. Which of the two equations has a solution?

Equations and trigonometric identities
17

Solve the equation 3 \sin \theta = 3 \sqrt{3} \cos \theta for 0 \lt \theta \lt 90.

18

Solve the following equations for 0 \degree \leq \theta < 360 \degree. Round your answers to the nearest degree if necessary.

a

2 \cos ^{2}\left(\theta\right) = 2 - \sin \theta

b

\sin ^{5}\left(\theta\right) \cos ^{3}\left(\theta\right) = 0

c

\cos ^{2}\left(\theta\right) - 8 \sin \theta \cos \theta + 3 = 0

d

\dfrac{1 - \tan ^{2}\left(\theta\right)}{1 + \tan ^{2}\left(\theta\right)} + \cos \theta = 0

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