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5.05 Solving trigonometric equations 1

Worksheet
Exact value equations
1

State whether the following equations have a solution:

a

\cos \theta - 4 = 0

b

9 \tan \theta + 4 = 0

2

State the number of solutions for \theta of the following equations in the domain \\ 0 \degree \lt \theta \lt 90 \degree.

a

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

b

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

c

\tan \theta = - 1

3

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

a

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

b

\tan \theta = \sqrt{3}

c

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

d

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

4

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

a

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

b

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

c

\cos \theta = 0

d

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

e

\sin \theta = 0

f

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

g

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

h

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

i

\sin \theta = 1

j

\tan \theta = \sqrt{3}

k

\tan \theta = 0

l

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

m

4 \tan \theta + 2 = - 2

n

8 \cos \theta - 4 = 0

o

2 \cos \theta + 4 = 3

p

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

Non-exact value equations
5
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

6

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

Different domains
7

State the number of solutions for the following equations in the domain -90 \degree \lt \theta \lt 90 \degree:

a

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

b

\cos \theta = - 0.914

8

State the number of solutions for the following equations in the domain 180 \degree \leq \theta \leq 360 \degree:

a

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

b

\sin \theta = - 0.697

9

Solve the following equations for the given domain:

a

\cos \theta = \dfrac{1}{\sqrt{2}} for 0 \degree \lt \theta \lt 90 \degree

b

\sin \theta = - \dfrac{1}{2} for - 90 \degree \leq \theta \leq 90 \degree

c

\tan \theta = - 1 for - 180 \degree \leq \theta \leq 0 \degree

d

\tan \theta = 3 - 2 \tan \theta for 0 \degree \leq \theta \leq 180 \degree

10

Solve the following equations for the domain - 180 \degree \leq \theta \leq 180 \degree. Round your answer to two decimal places if necessary.

a

\cos \theta = - 0.831

b

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

c

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

d

\sin \theta = 0.238

e

\tan \theta = 1

f

\tan \theta = 1.237

11

Solve the following equations for the domain 360 \degree \leq \theta \leq 720 \degree. Round your answer to two decimal places if necessary.

a

\sin \theta = 0.3584

b

\tan \theta = 2.2460

Equations with reciprocal functions
12

State the number of solutions for the equation \cot \theta = - 1 for 270 \degree \leq \theta \leq 360 \degree.

13

Solve the following equations for 0 \degree \leq \theta \leq 90 \degree. Round your answers to one decimal place.

a

\text{cosec } \theta = 2.645

b

\cot \theta = 2.867

14

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

a

\text{cosec } \theta = 1.6831

b

\text{cosec } \theta = - 1.7998

c

2 \left(\text{cosec } \theta + 1\right) = 12 - 5 \text{cosec } \theta

d

\text{cosec } \theta = 2

e

\sec \theta = 2.6711

f

\sec \theta = - 2.2925

g

6 \sec \theta - 1 = 3 \sec \theta - 7

h

\sec \theta = 2

i

\cot \theta = 0.3488

j

\cot \theta = - 1.1757

k

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

15

Solve the following equations for the given domain. Round your answers to the nearest minute if necessary.

a

\dfrac{4 \sqrt{3}}{3} - \cot \theta = 3 \cot \theta for 0 \degree \leq \theta \leq 180 \degree

b

\text{cosec } \theta = \dfrac{2}{\sqrt{3}} for - 90 \degree \leq \theta \leq 90 \degree

c

\sec \theta = 2 for - 180 \degree \leq \theta \leq 0 \degree

d

5 \cot \theta + 6 = 9 \cot \theta - 2 for \\ 0 \degree \leq \theta \leq 360 \degree

e

\sec \theta = 1.4543 for 0 \degree \lt \theta \lt 90 \degree

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Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

MA11-4

uses the concepts and techniques of periodic functions in the solutions of trigonometric equations or proof of trigonometric identities

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