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1.045 Trigonometric equations

Worksheet
Solve trigonometric equations graphically
1

Consider the function y = 3 \sin x and the line y = 3 which are graphed below:

-\frac{3}{2}π
-1π
-\frac{1}{2}π
\frac{1}{2}π
\frac{3}{2}π
x
-3
-2
-1
1
2
3
y

State all solutions to the equation 3 \sin x = 3 over the domain \left[ - 2 \pi , 2 \pi\right].

2
a

Graph the function y = \tan x and the line y = 1 on the same number plane over the domain \left[ - 2 \pi , 2 \pi\right].

b

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

3

Consider the graph of y = \cos x over the domain [0, 2 \pi ]:

\frac{1}{6}π
\frac{1}{3}π
\frac{1}{2}π
\frac{2}{3}π
\frac{5}{6}π
\frac{7}{6}π
\frac{4}{3}π
\frac{3}{2}π
\frac{5}{3}π
\frac{11}{6}π
x
-1.5
-1
-0.5
0.5
1
1.5
y
a

State the x-values for which \cos x = 0.

b

State the first x-value for which \cos x = 0.5

c

For what other value of x shown on the graph, does \cos x = 0.5?

d

For what values of x does \cos x = - 0.5?

4

Consider the graph of y = \tan x:

\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
\frac{5}{4}π
\frac{3}{2}π
\frac{7}{4}π
x
-3
-2
-1
1
2
3
y
a

How long is one period of the graph?

b

State the x-values for which \tan x = 0, from x = 0 to x = 2 \pi inclusive.

c

State the first x-value for which \tan x = 1.

d

For what other value of x shown on the graph does \tan x = 1?

e

For what values of x shown on the graph does \tan x = - 1?

5

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

a

Sketch the graph of this function over the domain [-4 \pi, 4\pi].

b

Sketch the line y = - 0.5 on the same number plane.

c

Hence, state all solutions to the equation \cos \left(\dfrac{x}{4}\right) = - 0.5 over the domain \left[ - 4 \pi , 4 \pi\right] in exact form.

6

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.

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 = \sin \left(x - \dfrac{\pi}{3}\right) + 5.

a

Sketch the function y = \sin \left(x - \dfrac{\pi}{3}\right) + 5 over the domain [- 2 \pi, 2 \pi].

b

Sketch the line y = \dfrac{11}{2} on your graph.

c

Hence, state all solutions to the equation \sin \left(x - \dfrac{\pi}{3}\right) + 5 = \dfrac{11}{2} over the domain \left[ - 2 \pi , 2 \pi\right). Give your answers in exact form.

9

Consider the equation 3 \sin \left( 3 x + \dfrac{\pi}{7}\right) = - \dfrac{11}{10}.

a

Which function would be graphed along with y = - \dfrac{11}{10} in order to solve the equation graphically?

b

Graph both of these functions using the graphing facility of your calculator. Hence state all solutions to the equation over the domain \left[ - \dfrac{13\pi}{42}, \dfrac{5\pi}{14}\right]. Round your answers correct to three decimal places.

10

Consider the equation - 5 \cos \left(\dfrac{x}{2} + \dfrac{\pi}{5}\right) = - \dfrac{17}{10}.

a

Which function would be graphed along with y = - \dfrac{17}{10} in order to solve the equation graphically?

b

Graph both of these functions using the graphing facility of your calculator. Hence state all solutions to the equation over the domain \left[ - \dfrac{7 \pi}{5} , \dfrac{13 \pi}{5}\right]. Round your answers correct to three decimal places.

Solve trigonometric equations algebraically
11

Explain whether the following equations have a solution:

a

\cos \theta - 4 = 0

b

2\tan \theta + 4 = 0

12

Solve the following equations for 0 \leq \theta \leq \dfrac{\pi}{2}:

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}

13

Solve the following equations for 0 \leq \theta \leq 2\pi :

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

14

Solve for x in the domain of 0 \leq x \leq 2 \pi:

a

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

b
\sqrt{3} \tan \left(\dfrac{x}{2}\right) = - 3
c

10 \sin x - 5 \sqrt{3} = 0

d

\cos \left(\dfrac{x}{2}\right) = 1 - \cos \left(\dfrac{x}{2}\right)

15

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
q

\sqrt{2} \cos \left(x - \dfrac{\pi}{3}\right) = 1 for - 2 \pi \leq x \leq 2 \pi

16

Solve \sin ^{2}\left(x\right) - 6 \cos ^{2}\left(x\right) = 1 over the interval [0,2 \pi).

17

Find the angle satisfying \cos \theta = 0.7482 for 0 < \theta < \dfrac{\pi}{2}. Round your answer correct to two decimal places.

18

Consider the equation \sin \theta = 0.2756.

a

Find the acute angle satisfying the equation. Round your answer to two decimal places.

b

Find the angles satisfying \sin \theta = 0.2756 for 2 \pi \leq \theta \leq 4 \pi. Round your answers to two decimal places.

19

Find the acute angle satisfying 2 \sin \theta + 3 = 6 \sin \theta. Round your answer to two decimal places.

20

Find the acute angle satisfying 5 \tan \theta + 4 = 9 \tan \theta - 1. Round your answer to three decimal places.

21

Find the angle satisfying \sin ^{2}\left(\theta\right) = 0.46 for 0 < \theta < \dfrac{\pi}{2}. Round your answer to two decimal places.

22

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?

23

Find the angles satisfying 12 \sin ^{2}\left(\theta\right) - 11 \sin \theta + 2 = 0 for 0 < \theta < \dfrac{\pi}{2}. Round your answers to three decimal places.

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3.2.3.3

use trigonometric functions and their derivatives to solve practical problems; including trigonometric functions of the form 𝑦 = sin(𝑓(𝑥)) and 𝑦 = cos(𝑓(𝑥)).

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