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7.02 Using radians

Worksheet
Radians
1

Calculate the following trigonometric ratios to two decimal places:

a

\sin \dfrac{35 \pi}{16}

b

\cos 6.87

c

\sin 7.26

d

\tan 7.26

e

\tan \left(\dfrac{- 3 \pi}{7}\right)

f

\cos \dfrac{2 \pi}{3}

g

\sin \left( - \dfrac{4 \pi}{3}\right)

h
\cos \dfrac{4 \pi}{5}
i
\sin \left( - \dfrac{4 \pi}{5} \right)
Exact values
2

Consider the following diagram:

a

Find the length of side h.

b

Hence, state the exact value of:

i
\sin \dfrac{\pi}{3}
ii
\sin \dfrac{\pi}{6}
iii
\tan \dfrac{\pi}{3}
iv
\cos \dfrac{\pi}{3}
v
\cot \dfrac{\pi}{6}
3

Consider the following diagram:

a

Find the length of the hypotenuse, h.

b

Hence, state the exact value of:

i
\sin \dfrac{\pi}{4}
ii
\cos \dfrac{\pi}{4}
iii
\tan \dfrac{\pi}{4}
iv
\cot \dfrac{\pi}{4}
v
\sec \dfrac{\pi}{4}
4

Consider the diagram of the unit circle:

Find the exact value of:

a
\sec \dfrac{\pi}{6}
b

\text{cosec } \dfrac{\pi}{4}

c

\cot \dfrac{\pi}{3}

5

Consider the unit circle diagram and state the exact value of the following trigonometric ratios:

a

\sin \dfrac{\pi}{2}

b

\cos \dfrac{3\pi}{2}

c

\tan \pi

d

\cos 0

e

\sec \dfrac{\pi}{2}

f

\sec \pi

g

\text{cosec } \dfrac{\pi}{2}

h

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

-1
1
0
-1
1
\dfrac{\pi}{2}
6

Find the exact value of the following:

a

\sin \dfrac{\pi}{3} + \cos \dfrac{\pi}{3}

b

\sin \dfrac{\pi}{6} \cos \dfrac{\pi}{4}

c

\dfrac{\sin \dfrac{\pi}{3}}{\cos \dfrac{\pi}{6}}

d

\sin \dfrac{\pi}{4} \cos \dfrac{\pi}{6} + \tan \dfrac{\pi}{4}

e

\sin ^{2}\left(\dfrac{\pi}{6}\right) - \cos ^{2}\left(\dfrac{\pi}{3}\right)

f

2\sin ^{2}\left(\dfrac{\pi}{2}\right) + 3\cos ^{2}\left(\dfrac{\pi}{2}\right)

Exact values from reference angles
7

Consider the unit circle shown, where points A and B have the same \\ y-coordinates.

Suppose that \theta = \dfrac{10 \pi}{11}. State the size of the reference angle, \alpha.

8

Consider the unit circle shown, where the line through A and B passes through the origin, O.

Suppose that \theta = \dfrac{8 \pi}{7}. State the size of the reference angle, \alpha.

9

Consider the unit circle shown, where the points A and B have the same \\ x-coordinate.

Suppose that \theta = \dfrac{9 \pi}{5}. State the size of the reference angle, \alpha.

10

Find the exact value of the following:

a

\sin \dfrac{5 \pi}{6}

b

\tan \dfrac{3 \pi}{4}

c

\sin \dfrac{7 \pi}{6}

d

\cos \dfrac{7 \pi}{6}

e

\sin \dfrac{5 \pi}{3}

f

\cos \dfrac{5 \pi}{3}

g

\cos 4 \pi

h

\tan 9 \pi

i

\sin \dfrac{ 5\pi}{2}

j

\cos \dfrac{ 7\pi}{2}

k

\cos \dfrac{3 \pi}{4}

l
\sin \dfrac{5 \pi}{4}
m
\cos \dfrac{5 \pi}{4}
n
\tan \dfrac{5 \pi}{4}
o

\tan \dfrac{7 \pi}{6}

p

\tan \dfrac{11 \pi}{6}

11

Find the exact value of the following:

a

\sin \left( - \dfrac{17 \pi}{6} \right)

b

\cos \left( - \dfrac{17 \pi}{6} \right)

c

\cos \left( - \dfrac{4 \pi}{3} \right)

d

\tan \left( - \dfrac{17 \pi}{6} \right)

e

\text{cosec } \left( - \dfrac{17 \pi}{6} \right)

f

\sec \left( - \dfrac{17 \pi}{6} \right)

g

\cot \left( - \dfrac{17 \pi}{6} \right)

h

\cot 3 \pi

i
\text{cosec} \dfrac{5 \pi}{4}
j
\sec \dfrac{5 \pi}{4}
k
\cot \dfrac{5 \pi}{4}
12

Find the exact value of the following:

a

\dfrac{\left(\sin \dfrac{2 \pi}{3}\right) \left(\cos \dfrac{2 \pi}{3}\right) \left(\tan \dfrac{3 \pi}{4}\right)}{\tan \left( - \dfrac{\pi}{4} \right)}

b

\dfrac{\sin \dfrac{2 \pi}{3} + \cos \dfrac{5 \pi}{6} - \tan \dfrac{7 \pi}{4}}{\cos \dfrac{4 \pi}{3}}

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Outcomes

MA11-3

uses the concepts and techniques of trigonometry in the solution of equations and problems involving geometric shapes

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