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Standard Level

5.09 Solving equations using radians

Worksheet
Exact values
1

Consider the equation \sin \theta = - \dfrac{1}{2}. State the number of solutions for \theta, in the domain \\ 0 \lt \theta \lt \dfrac{\pi}{2}.

2

For each of the following equations, find the exact value of acute angle x, in the domain \\ 0 \leq x \leq \dfrac{\pi}{2}:

a

\tan x = 1

b

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

c

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

d

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

3

For each of the following, find the exact values of x, where - \pi \lt x \lt \pi:

a

\tan x = \sqrt{3}

b

\tan x = \dfrac{1}{\sqrt{3}}

c

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

d

\tan \left(x - \dfrac{2 \pi}{3}\right) = 0

4

For each of the following, find the exact values of \theta, in the given domain:

a

\tan \theta = 0, where - \pi \leq \theta \leq \pi

b

\sin \theta = 0, where - 4 \pi \leq \theta \leq 4 \pi

c

12 \cos \theta - 6 \sqrt{3} = 0, where - \pi \lt \theta \lt \pi

d

4 \sin \theta - 3 = - 5, where - 2\pi \leq \theta \leq 2\pi

e

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

f

4 \sin ^{2}\left(\theta\right) = 1, where - 4 \pi \lt \theta \lt 4 \pi

5

For each of the following, find the exact values of x, over the domain \left[ 0 , 2 \pi \right):

a

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

b

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

c

6 \cos x + 2 = - 1

d

6 \cos x - 3 \sqrt{2} = 0

e

2 \tan x + 3 = 5

f

2 \sin x + 2 = 1

g

\cos x \tan x = \cos x

h

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

i

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

j

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

k

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

l

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

m

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

n

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

o

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

p

\sin ^{2}\left(\dfrac{x}{2}\right) - 1 = 0

q

36 \left(1 - \cos x\right) \left(1 + \cos x\right) = 27

6

For each of the following, find the exact values of x, over the domain 0 \leq x \leq 2 \pi:

a

\sin x = 1

b

\cos x = - 1

c

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

d

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

e

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

f

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

g

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

h

4 \cos x - 2 = 0

i

2 \sqrt{3} \tan x - 2 = 0

j

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

k

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

l

\sin \left(x + \dfrac{\pi}{6}\right) = \dfrac{1}{\sqrt{2}}

m

\cos \left(x + \dfrac{\pi}{6}\right) = - \dfrac{1}{2}

n

\sin x = - \cos x

o

\sin \left(x + \dfrac{\pi}{3}\right) = \dfrac{1}{2}

p

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

q

\cos \left(x + \dfrac{\pi}{4}\right) = - \dfrac{1}{\sqrt{2}}

r

\sin \left(x - \dfrac{4 \pi}{3}\right) = \dfrac{1}{2}

7

Find the exact solutions of the following equations where 0 \leq \theta \leq 2 \pi:

a
\cos^{2} \theta - 5 \cos \theta + 4 = 0
b
2\sin^{2} \theta + \sin \theta -1 = 0
c
\tan^{2} \theta + \sqrt{3} \tan \theta = 0
d
\sin^{2} \theta - \cos^{2} \theta = 0
Other radian values
8

Consider the equation \cos \theta = 0.42. State the number of solutions for \theta over the domain \\ 0 \lt \theta \lt 2\pi.

9

For each of the following equations:

i

Find the acute angle \theta in radians that solves the equation, correct to two decimal places.

ii
Hence find all solutions to \theta over the domain \left[0,2 \pi \right], correct to two decimal places.
a

\sin \theta = 0.3420

b

\cos \theta = 0.9063

c

\tan \theta = 0.8340

d

\cos \theta = 0.2345

e

\sin \theta = 0.9921

f

\tan \theta = 0.7743

g

\cos \theta = 0.67

h

\sin \theta = 0.091

i

\cos 2 \theta = 0.9

10

Jessica finds the acute angle solution to the equation \sin x = k, is x = p.

Find another solution for the equation over the domain 0 \lt x \lt 2\pi, in terms of p.

11

Toby finds the acute angle solution to the equation \cos x = m, is x = n.

Find another solution for the equation over the domain 0 \lt x \lt 2\pi, in terms of n.

12

Sarah finds the acute angle solution to the equation \tan x = h, is x = t.

Find another solution for the equation over the domain 0 \lt x \lt 2\pi, in terms of t.

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