Topics
2. Trigonometric Functions and Graphs
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AustraliaNSW
Year 12

2.03 Solutions to trigonometric equations

Lesson
Worksheet
Practice
Worksheet
Algebraic solutions
1

Solve each equation for the given interval. Give your answers in exact form.

a
\sin \left(x - \dfrac{\pi}{4}\right) = \dfrac{1}{\sqrt{2}} for - \pi \leq x < \pi
b
\cos \left(x + \dfrac{\pi}{4}\right) = - \dfrac{1}{\sqrt{2}} for 0 \leq x< 2\pi
c
\cos \left(x - \dfrac{\pi}{5}\right) = \dfrac{\sqrt{3}}{2} for 0 \leq x \leq 3 \pi
d
\sqrt{2} \cos \left(x - \dfrac{\pi}{3}\right) = 1 for- 2 \pi \leq x \leq 2 \pi
e
\tan \left(x + \dfrac{\pi}{5}\right) = \sqrt{3} for 0 \leq x \leq 2 \pi
f
\sin \left( 3 x - \dfrac{\pi}{3}\right) = - 1 for - \pi \leq x \leq \pi
g
2 \sin \left(\dfrac{x}{2} + \dfrac{\pi}{5}\right) = 1 for - 4 \pi \leq x \leq 4 \pi
h
\tan \left( 2 x - \dfrac{\pi}{6}\right) = - \dfrac{1}{\sqrt{3}} for 0\leq x \leq \pi
i
\tan \left( \pi x - \dfrac{\pi}{4}\right) = \dfrac{1}{\sqrt{3}} for - 2 \leq x \leq 2
j
2 \tan \left( 4 x + \dfrac{\pi}{5}\right) = 1 for - \dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}
k
2\sin (3x + \dfrac{3\pi}{2}) + 1 = 0 for - \dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}
l
6 \cos(3x - \dfrac{\pi}{2}) + 3 = 0 for - \dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}
m
2\tan (2x + \dfrac{\pi}{9}) + \dfrac{2}{\sqrt{3}} = 0 for 0 \leq x \leq \pi
n
-2 \cos(3x + \dfrac{\pi}{3}) + 1 = 0 for 0 \leq x \leq \pi
o
\sin ^2(x - \dfrac{\pi}{4}) - 1=0 for 0 \leq x \leq 2\pi
p
12\cos ^2(2x - \dfrac{\pi}{6}) - 3=0 for 0 \leq x \leq \pi
Graphical solutions
2

Consider the function y = \cos \left(x - \dfrac{\pi}{6}\right).

a

Sketch the graph of the function for -2\pi \leq x \leq 2\pi.

b

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

c

Hence, state all solutions to the equation \cos \left(x - \dfrac{\pi}{6}\right) = \dfrac{1}{2} over the domain \left( - 2 \pi , 2 \pi\right]. Give your answers as exact values.

3

Consider the function y = \sin \left(x - \dfrac{\pi}{3}\right).

a

Sketch the graph of the function for -2\pi \leq x \leq 2\pi..

b

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

c

Hence, state all solutions to the equation \sin \left(x - \dfrac{\pi}{3}\right) = \dfrac{1}{2} over the domain \left[ - 2 \pi , 2 \pi\right). Give your answers as exact values.

4

Consider the function y = \tan \left(x - \dfrac{\pi}{4}\right).

a

Sketch the graph of the function for -2\pi \leq x \leq 2\pi.

b

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

c

Hence, state all solutions to the equation \tan \left(x - \dfrac{\pi}{4}\right) = 1 over the domain \left[ - 2 \pi , 2 \pi\right). Give your answers as exact values.

5

Consider the function y = 2 \sin 4 x.

a

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

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[ - 90 \degree , 90 \degree\right]. Give your answers in degrees.

6

Consider the function y = 2 \sin 2 x.

a

Sketch the graph of the function for -180\degree \leq x \leq 180\degree.

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]. Give your answers in degrees.

7

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

a

Sketch the graph of the function for -\pi \leq x \leq \pi.

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[ - \pi , \pi\right]. Give your answers as exact values.

8

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

a

Sketch the graph of the function for -\dfrac{2\pi}{3} \leq x \leq \dfrac{2\pi}{3}.

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[ - \dfrac{2 \pi}{3} , \dfrac{2 \pi}{3}\right]. Give your answers as exact values.

9

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

a

Sketch the graph of the function for -120\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[ - 120 \degree , 120 \degree\right]. Give your answers in degrees.

Technology
10

Solve each equation for the given interval. Round your answers to two decimal places.

a
\cos 2 x = 0.9 for 0 \leq x < 2 \pi
b
3\sin 2 x = 1.2 for 0 \leq x < \pi
c
\tan \left(\dfrac{x}{2}\right) = 5 for -3\pi \leq x < 3\pi
d
4\cos\left(\dfrac{x}{3}+2\right) = 1 for 0 \leq x < 4 \pi
11

Consider the equation \sin \left(\dfrac{x}{2} + 60 \degree\right) = \cos \left(\dfrac{x}{2} - 60 \degree\right).

a

State the two functions you would graph in order to solve this equation graphically.

b

Sketch the graph of both of these functions using technology.

c

Hence, state all solutions to the equation over the domain \left[ - 360 \degree, 360 \degree\right].

12

Use technology to solve the following functions over the interval [0 \degree, 360 \degree). Give all solutions to the nearest degree.

a
2 \sin 3 x - 4 \sin 2 x = 0
b
2 \sin x + 5 \cos x = 1
c
\cos 3 x + \cos x = \sin x
d
2 \cos 3 x - 3 \cos 2 x = 0
e
\cos \left(\dfrac{x}{2}\right) - 4 \sin 2 x = 0
f
\sin \left(\dfrac{x}{2}\right) = 7 \cos 3 x
g
4 \sin ^{5}\left(x\right) = - \left( \cos x + 1 \right)
h
\cos ^{3}\left(x\right) + \cos x = - 1
i
\sin 3 x = \ln 3 x
j
e^{x} - 2 = \cos x
13

Use technology to solve the following equations for 0 \leq x \leq 2\pi. Give all solutions to three decimal places.

a
2 \sin x + 5 \cos x = 1
b
3 \sin 2 x + 2 \sin 3 x = 0
c
3\sin(2x)=x+1
d
2 \cos 3 x + 3 \cos 2 x = 0
e
\cos 3 x + \cos x = \sin x
f
\cos \left(\dfrac{x}{2}\right) - 4 \sin 2 x = 0
g
\sin \left(\dfrac{x}{2}\right) = 5 \cos 3 x
h
2\tan(x - \dfrac{\pi}{3}) = -\dfrac{x}{2}
i
4 \sin ^{5}\left(x\right) = - \left( \cos x + 1 \right)
j
\cos ^{5}\left(x\right) + \cos x = - 1
k
e^{x} - 6 = \cos x
l
\sin 2 x = \ln 2 x
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Outcomes

MA12-1

uses detailed algebraic and graphical techniques to critically construct, model and evaluate arguments in a range of familiar and unfamiliar contexts

MA12-5

applies the concepts and techniques of periodic functions in the solution of problems involving trigonometric graphs

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