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7.01 Exact ratios and the unit circle

Worksheet
Exact value triangles
1

Find the exact value of the following:

a

\sin 30 \degree

b

\sin 60 \degree

c

\sin 45 \degree

d

\cos 45 \degree

e

\cos 60 \degree

f

\cos 30 \degree

g

\tan 60 \degree

h

\tan 45 \degree

i

\tan 30 \degree

2

Evaluate the following expressions, leaving your answer in exact, rationalised form:

a
\dfrac{\sin 60 \degree}{\cos 30 \degree}
b
\sin 45 \degree \cos 30 \degree
c
\sin 45 \degree + \cos 30 \degree
d
\sin 45 \degree \cos 60 \degree + \tan 45 \degree
Unknown sides
3

Find the exact value of the pronumeral in the following triangles:

a
b
c
d
4

Consider the given triangle:

a

Find the exact length of side a.

b

Find the exact length of side b.

5

For each of the following triangles:

i

Find the exact value of x.

ii

Find the exact value of y.

a
b
6

Consider the unknown lengths in the given figure:

a

Find m.

b

Find t.

Unknown angles
7

\theta is an angle in a right-angled triangle. Find the value of \theta in the following equations:

a

\cos \theta = \dfrac{1}{2}

b

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

c

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

d

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

8

Find the value of \theta if \cos \theta = \dfrac{1}{\sqrt{2}} and \sin \theta = \dfrac{1}{\sqrt{2}}.

9

Given that \cos \theta = \dfrac{\sqrt{3}}{2} and \sin \theta = \dfrac{1}{2}:

a

Find the value of \theta.

b

Find the value of \tan \theta.

10

Find the unknown \theta in the following triangles:

a
b
Unit circle
11

State the quadrant in which the following angles are located:

a

299 \degree

b

5 \degree

c

160\degree

d

229\degree

e

40\degree

f

310\degree

g

138\degree

h

344\degree

12

Write four different angles between 0 \degree and 360 \degree inclusive, that lie on the quadrant boundaries.

13

State the quadrant where the angle in each scenario is located:

a

\theta is an angle such that \sin \theta > 0 and \cos \theta < 0.

b

\theta is an angle such that \tan \theta < 0 and \sin \theta > 0.

c

\theta is an angle such that \tan \theta < 0 and \cos \theta < 0.

d

\theta is an angle such that \tan \theta > 0 and \sin \theta > 0.

14

State whether the values of the following are positive or negative:

a

\sin 31 \degree

b

\tan 31 \degree

c

\cos 267 \degree

d

\sin 267 \degree

e

\cos 180 \degree

f

\tan 296 \degree

g

\sin 120 \degree

h

\cos 91 \degree

i

\sin 296 \degree

j

\cos 120 \degree

k

\cos 296 \degree

l

\sin 90 \degree

m

\cos 51 \degree

n

\sin 51 \degree

o

\cos 233 \degree

p

\tan 233 \degree

15

Consider the point \left(x, y\right) on the following unit circles:

i

Find the value of x.

ii

Find the value of y.

a
b
16

Consider the point on the given unit circle. Find the value of \theta.

17

The first diagram shows a unit circle with point P \left(\dfrac{1}{\sqrt2}, \dfrac{1}{\sqrt2}\right) marked on the circle. Point P represents a rotation of 45 \degree anticlockwise around the origin from the positive x-axis:

a

Find the exact values of the following:

i

\sin 45\degree

ii

\cos 45\degree

iii

\tan 45\degree

b

On the second diagram, the coordinate axes shows a 45 \degree angle that has also been marked in the second, third, and fourth quadrants. For each quadrant, find the relative angle.

i

Quadrant 2

ii

Quadrant 3

iii

Quadrant 4

c

The points Q, R and S mark rotations of point P to the corresponding angles on the unit circle. State the exact coordinates of each point:

i

Q

ii

R

iii

S

d

Write the following in terms of an equivalent ratio of 45 \degree:

i

\sin 135\degree

ii

\cos 225 \degree

iii

\tan 315 \degree

e

Hence find the exact value of the following:

i

\sin 135\degree

ii

\cos 225 \degree

iii

\tan 315 \degree

18

The first diagram shows a unit circle with point P \left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) marked on the circle. Point P represents a rotation of 60 \degree anticlockwise around the origin from the positive x-axis:

a

Find the exact values of the following:

i

\sin 60\degree

ii

\cos 60\degree

iii

\tan 60\degree

b

On the second diagram, the coordinate axes shows a 60 \degree angle that has also been marked in the second, third, and fourth quadrants. For each quadrant, find the relative angle.

i

Quadrant 2

ii

Quadrant 3

iii

Quadrant 4

c

The points Q, R and S mark rotations of point P to the corresponding angles on the unit circle. State the exact coordinates of each point:

i

Q

ii

R

iii

S

d

Write the following in terms of an equivalent ratio of 60 \degree:

i

\sin 120\degree

ii

\cos 240 \degree

iii

\tan 300 \degree

e

Hence find the exact value of the following:

i

\sin 120\degree

ii

\cos 240 \degree

iii

\tan 300 \degree

19

The first diagram shows a unit circle with point P \left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) marked on the circle. Point P represents a rotation of 30 \degree anticlockwise around the origin from the positive x-axis:

a

Find the exact values of the following:

i

\sin 30\degree

ii

\cos 30\degree

iii

\tan 30\degree

b

On the second diagram, the coordinate axes shows a 30 \degree angle that has also been marked in the second, third and fourth quadrants. For each quadrant, find the relative angle:

i

Quadrant 2

ii

Quadrant 3

iii

Quadrant 4

c

The points Q, R and S mark rotations of point P to the corresponding angles on the unit circle. State the exact coordinates of each point:

i

Q

ii

R

iii

S

d

Write the following in terms of an equivalent ratio of 30 \degree:

i

\cos 150\degree

ii

\sin 210 \degree

iii

\tan 330 \degree

e

Hence find the exact value of the following:

i

\cos 150\degree

ii

\sin 210 \degree

iii

\tan 330\degree

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Outcomes

VCMMG368 (10a)

Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies.

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