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VCE 11 Methods 2023

8.02 Degrees and Radians

Worksheet
Unit circle
1

The diagram shows a number of points on the unit circle:

State the point that corresponds to a distance of \dfrac{\pi}{2} units around the circle.

-1
1
x
-1
1
y
2

The diagram shows a number of points on the unit circle:

State the point that corresponds to a distance of \dfrac{21 \pi}{4} units around the circle.

-1
1
x
-1
1
y
3

The diagram shows a number of points on the unit circle:

State the point that is closest to a distance of 1.6 radians around the circle.

-1
1
x
-1
1
y
4

What fraction of the circumference of the unit circle do the following angles represent?

a

\dfrac{\pi}{4}

b

\dfrac{4 \pi}{7}

c

5 \pi

d

\dfrac{7 \pi}{2}

5

The diagram shows the circumference of a unit circle divided into 12 equal arcs, and a number of points on the unit circle.

State two exact angles between - 2 \pi and 2 \pi that correspond to the following points:

a

S

b

R

c

P

d

Q

x
y
Convert between degrees and radians
6
a

State the multiplier to convert an angle from degrees to radians.

b

State the multiplier to convert an angle from radians to degrees.

7

Convert the following to radians, giving your answers in exact form:

a

180 \degree

b

360 \degree

c

90 \degree

d

30 \degree

e

60 \degree

f

210 \degree

g

45 \degree

h

225 \degree

i

112 \degree

j

- 300 \degree

k

- 270 \degree

l

- 135 \degree

8

Convert the following to radians, correct to two decimal places:

a

29 \degree

b

161.17 \degree

c

321 \degree 31 '

d

234 \degree 09 '

9

Convert the following angles in radians to degrees, correct to one decimal place when necessary:

a

\dfrac{\pi}{3}

b

\dfrac{2 \pi}{3}

c

- \dfrac{5 \pi}{3}

d

4.2

Trigonometric ratios
10

Consider the location of the angle on the unit circle and hence state the exact values of the following:

a

\sin \pi

b

\sin \dfrac{\pi}{2}

c

\tan 9 \pi

d

\cos 4 \pi

11

Consider the trigonometric ratio \sin \dfrac{5 \pi}{6}.

a

Determine the quadrant in which \dfrac{5 \pi}{6} is located.

b

Hence state whether \sin \dfrac{5 \pi}{6} is positive or negative.

c

Find the positive acute angle that \dfrac{5 \pi}{6} is related to.

d

Hence rewrite \sin \dfrac{5 \pi}{6} in terms of its related acute angle.

12

Consider the trigonometric ratio \cos \dfrac{5 \pi}{4}.

a

Determine the quadrant in which \dfrac{5 \pi}{4} is located.

b

Hence state whether \cos \dfrac{5 \pi}{4} is positive or negative.

c

Find the positive acute angle that \dfrac{5 \pi}{4} is related to.

d

Hence rewrite \cos \dfrac{5 \pi}{4} in terms of its related acute angle.

13

Consider the trigonometric ratio \tan \dfrac{5 \pi}{3}.

a

Determine the quadrant in which \dfrac{5 \pi}{3} is located.

b

Hence state whether \tan \dfrac{5 \pi}{3} is positive or negative.

c

Find the positive acute angle that \dfrac{5 \pi}{3} is related to.

d

Hence rewrite \tan \dfrac{5 \pi}{3} in terms of its related acute angle.

14

Consider the angle \dfrac{2 \pi}{3}.

a

Determine the quadrant in which the angle is located.

b

Express the following ratios in terms of a related acute angle:

i
\sin\dfrac{2 \pi}{3}
ii
\cos\dfrac{2 \pi}{3}
iii
\tan\dfrac{2 \pi}{3}
15

Consider the angle \dfrac{7 \pi}{6}.

a

Determine the quadrant in which the angle is located.

b

Express the following ratios in terms of a related acute angle:

i
\sin\dfrac{7 \pi}{6}
ii
\cos\dfrac{7 \pi}{6}
iii
\tan\dfrac{7 \pi}{6}
16

Using the approximations \cos \dfrac{\pi}{3} = 0.50, and \sin \dfrac{\pi}{3} = 0.87, write down the approximate value of the following, correct to two decimal places:

a

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

b

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

c

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

d

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

17

Using the approximations \cos \dfrac{\pi}{5} = 0.81, and \sin \dfrac{\pi}{5} = 0.59, write down the approximate value of the following, correct to two decimal places:

a
\cos \dfrac{4 \pi}{5}
b
\cos \dfrac{6 \pi}{5}
c
\sin \dfrac{6 \pi}{5}
d
\sin \left( - \dfrac{4 \pi}{5} \right)
18

Suppose s is a real number that corresponds to the point \left( - \dfrac{5}{13} , \dfrac{12}{13}\right) on the unit circle:

a

State the exact value of \sin \left(s + 6 \pi\right).

b

State the exact value of \cos \left(s + 6 \pi\right).

19

Suppose p is a real number that corresponds to the point \left( - \dfrac{4}{5} , \dfrac{3}{5}\right) on the unit circle:

a

Find the exact value of \sin \left( - p \right).

b

Find the exact value of \cos \left( - p \right).

20

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

a

Find the exact value of \sin \left(q + \pi\right).

b

Find the exact value of \cos \left(q + \pi\right).

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Outcomes

U2.AoS1.1

the unit circle, radians, arc length and sine, cosine and tangent as functions of a real variable

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