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Middle Years

6.07 Trigonometric equations

Worksheet
Graphical solutions
1

Consider the function y = \cos \left(x - 30\right).

a

Sketch the graph of the function for 0 \leq x \leq 360.

b

Sketch the line y = \dfrac{1}{2} on the same number plane.

c

Hence, state all solutions to the equation \cos \left(x - 30\right) = \dfrac{1}{2} over the domain \left( 0 , 360\right].

2

Consider the function y = \sin \left(x - 60\right).

a

Sketch the graph of the function for 0 \leq x \leq 360.

b

Sketch the line y = \dfrac{1}{2} on the same number plane.

c

Hence, state all solutions to the equation \sin \left(x - 60\right) = \dfrac{1}{2} over the domain \left[0, 360\right).

3

Consider the function y = 2 \sin 4 x.

a

Sketch the graph of the function for 0\leq x \leq 120.

b

Sketch the line y = 1 on the same number plane.

c

Hence, state all solutions to the equation 2 \sin 4 x = 1 over the domain \left[ 0 , 90 \right]. Give your answers in degrees.

4

Consider the function y = 2 \sin 2 x.

a

Sketch the graph of the function for 0 \leq x \leq 180.

b

State the other function you would add to the graph in order to solve the equation 2 \sin 2 x = 1.

c

Sketch the graph of this function on the same number plane.

d

Hence, state all solutions to the equation 2 \sin 2 x = 1 over the domain \left[ - 180 \degree , 180 \degree\right].

5

Consider the function y = 3 \cos 2 x + 1.

a

Sketch the graph of the function for 0 \leq x \leq 180.

b

State the other function you would add to the graph in order to solve the equation 3 \cos 2 x + 1 = \dfrac{5}{2}.

c

Sketch the graph of this function on the same number plane.

d

Hence, state all solutions to the equation 3 \cos 2 x + 1 = \dfrac{5}{2} over the domain \left[ 0 , 180\right].

6

Consider the function y = 2 \sin 3 x - 3.

a

Sketch the graph of the function for 0 \leq x \leq 60.

b

State the other function you would add to the graph in order to solve the equation 2 \sin 3 x - 3 = - 2.

c

Sketch the graph of this fuction on the same number plane.

d

Hence, state all solutions to the equation 2 \sin 3 x - 3 = - 2 over the domain \left[0 , 60\right].

7

Consider the function y = - 2 \cos 3 x.

a

Sketch the graph of the function for 0\degree \leq x \leq 120\degree.

b

State the other function you would add to the graph in order to solve the equation - 2 \cos 3 x = -1.

c

Sketch the graph of this function on the same number plane.

d

Hence, state all solutions to the equation - 2 \cos 3 x = -1 over the domain \left[ 0 \degree , 120 \degree\right].

Exact value equations
8

State whether the following equations have a solution:

a

\cos \theta - 4 = 0

b

9 \tan \theta + 4 = 0

9

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

10

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}

11

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
12
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

13

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

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