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5.085 Trigonometric identities

Worksheet
Unit circle symmetries
1

Let \theta be an acute angle in radians. If \sin \theta = 0.6, find the value of the following:

a

\sin \left(\pi - \theta\right)

b

\sin \left(\pi + \theta\right)

c

\sin \left( 2 \pi - \theta\right)

d

\sin \left( - \theta \right)

2

Let \theta be an acute angle in radians. If \cos \theta = 0.1, find the value of the following:

a

\cos \left(\pi - \theta\right)

b

\cos \left(\pi + \theta\right)

c

\cos \left( 2 \pi - \theta\right)

d

\cos \left( - \theta \right)

3

Let \theta be an acute angle in radians. If \tan \theta = 0.52, find the value of the following:

a

\tan \left(\pi - \theta\right)

b

\tan \left(\pi + \theta\right)

c

\tan \left( 2 \pi - \theta\right)

d

\tan \left( - \theta \right)

4

If \sin \dfrac{2 \pi}{11} = 0.5406, find the value of \sin \dfrac{9 \pi}{11} correct to four decimal places.

5

If \cos \dfrac{2 \pi}{9} = 0.7660, find the value of \cos \dfrac{7 \pi}{9} correct to four decimal places.

6

If \tan \dfrac{3 \pi}{7} = 4.3813, find the value of \tan \dfrac{4 \pi}{7} correct to four decimal places.

7

Suppose s is a real number that corresponds to the point \left( - \dfrac{8}{17} , \dfrac{15}{17}\right) on the unit circle:

a

Find the coordinates of \left(s - \dfrac{\pi}{2}\right).

b

Find the value of \sin \left(s - \dfrac{\pi}{2}\right).

c

Find the value of \cos \left(s - \dfrac{\pi}{2}\right).

8

Suppose s is a real number that corresponds to the point \left( - \dfrac{3}{13} , \dfrac{4\sqrt{10}}{13}\right) on the unit circle:

a

Find the coordinates of \left(s + \dfrac{\pi}{2}\right).

b

Find the value of \sin \left(s + \dfrac{\pi}{2}\right).

c

Find the value of \cos \left(s + \dfrac{\pi}{2}\right).

Complementary angles
9

For each of the following acute angles, state the complementary angle:

a
34\degree
b
62\degree
c
\dfrac{\pi}{6}
d
\dfrac{2\pi}{7}
10

Consider the following right triangle:

a

Find an expression for the following:

i

\cos \theta

ii

\sin \left(90 \degree - \theta\right)

iii

\sin \theta

iv

\cos \left(90 \degree - \theta\right)

b

Describe the rule found in part (a) using words.

c

If \sin \alpha = 0.32, find \cos\left(90\degree-\alpha\right).

d

If \sin \alpha = \cos \beta, find \alpha + \beta.

11

Find the acute angle \theta in the following equations:

a
\sin \theta = \cos 25 \degree
b
\cos \theta = \sin 85 \degree
c
\cos \theta = \sin \dfrac{\pi}{8}
d
\sin\dfrac{2\pi}{5} = \cos\theta
12

Given that \sin x = 0.19, find the exact value of \cos \left(\dfrac{\pi}{2} - x\right).

13

Find the exact value of c, if \sin \left(\dfrac{\pi}{9} + c\right) = \cos \left(\dfrac{\pi}{4}\right).

14

Find the value of x in the following equations:

a

\sin \left( 5 x + 40 \degree\right) = \cos \left( 3 x + 10 \degree\right)

b

\sin \left( 6 x + \dfrac{\pi}{3}\right) = \cos \left( 4 x + \dfrac{\pi}{9}\right)

15

Simplify:

a
\sin \left(90 \degree - p\right)
b

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

c
\dfrac{\cos 26 \degree}{7 \sin 64 \degree}
d

\sin \left(90 \degree - y\right) \times \tan y

16

Prove that \dfrac{\sin x \cos \left(\dfrac{\pi}{2} - x\right)}{\cos x \sin \left(\dfrac{\pi}{2} - x\right)} = \tan ^{2}\left(x\right).

Pythagorean identity
17

State the exact value of the following:

a

\sin ^{2}\left(20 \degree\right) + \cos ^{2}\left(20 \degree\right)

b

\sin ^{2}\left(\dfrac{\pi}{5}\right) + \cos ^{2}\left(\dfrac{\pi}{5}\right)

18

Given that \cos x = \dfrac{12}{13} where x is in the first quadrant:

a

Find the exact value of \sin x.

b

Find the exact value of \tan x.

c

Prove the following:

i

\sin ^{2}x = 1- \cos ^{2}x

ii
\tan x=\dfrac{\sin x}{\cos x}
19

Given that \sin \theta = \dfrac{\sqrt{3}}{2}, where 90 \degree < \theta < 180 \degree:

a

In which quadrant does angle \theta lie?

b

Find the value of \cos \theta.

20

Given that \cos y = - \dfrac{5}{13}, where 180 \degree < y < 360 \degree:

a

In which quadrant does angle y lie?

b

Find the value of \tan y.

21

Simplify the following expressions:

a

\tan \theta \cos \theta

b

\left(\cos \theta - \sin \theta\right)^{2}

c

\dfrac{1 - \cos ^{2}\left(\theta\right)}{1 - \sin ^{2}\left(\theta\right)}

d

\dfrac{1}{1 - \cos \theta} + \dfrac{1}{1 + \cos \theta}

22

Prove the following identities:

a

\dfrac{\sin x}{\cos x \tan x} = 1

b

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

c

\dfrac{\sin ^{2}\left(x\right) + \sin x \cos x}{\cos ^{2}\left(x\right) + \sin x \cos x} = \tan x

d

\dfrac{\sin \theta}{1 - \cos \theta} = \dfrac{1 + \cos \theta}{\sin \theta}

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Outcomes

1.2.14

prove and apply the angle sum and difference identities

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