Angles and the Unit Circle

Section 13.2

An angle is in standard position when the vertex

is at the origin and one ray is on the positive x-

axis. The ray on the x-axis is the initial side of the

angle; the other ray is the terminal side of the

angle.

To measure an angle in standard position, find the

amount of rotation from the initial side to the

terminal side.

x

y

origin

initial side

term

inal

sid

e

Standard Position

One full rotation contains 360 degrees.

1. How many degrees are in one quarter of a

rotation?

90°

2. How many degrees in one half of a rotation?

180°

3. How many in three quarters of a rotation?

270°

360°

90°

180°

270°

x

y

x

y

x

y

x

y

x

y

0° 360°

Example 1

a. Find the measure of the angle below.

180° + 20° = 200°

x

y

20°

b. Find the measure of the angle below.

x

y

30°

150°

The measure of an angle is positive when the

rotation from the initial side to the terminal side is

in the counterclockwise direction.

The measure is negative when the rotation is

clockwise.

x

y

+120°

x

y

−120°

Example 2

Sketch each angle in standard position.

a. 36°

x

y

36°

b. 315°

x

y

315°

x

yc. −150°

−150°

Two angles in standard position are coterminal

angles if they have the same terminal side.

x

y

135°

−225°

These angles

are coterminal

angles since

they both have

the same

terminal side.

Example 3

Find the measure of an angle between 0° and

360° coterminal with each given angle.

a. −100°

To find a coterminal angle for a negative

angle add 360° until you get an angle

between 0° and 360°.

−100° + 360° = 260°

b. 480°

For angle that is greater than 360° subtract

360° from the given angle until you get an

angle between 0° and 360°.

480° − 360° = 120°

Unit Circle

The unit circle has a radius of 1 unit and its

center is at the origin of the coordinate plane.

radius

unit

circle We will call the

measure of an angle

in standard position,

θ.

Cosine and Sine of an Angle Definition

We will find the coordinates of points on the unit

circle using the special triangles that you learned

in Geometry.

Suppose an angle in standard position has a

measure θ.

The cosine of θ (cos θ) is the x-coordinate of the

point at which the terminal side of the angle

intersects the unit circle.

The sine of θ (sin θ) is the y-coordinate.

30°

P(cos 30°, sin 30°)

3 1,

2 2

2 2,

2 2

1 3,

2 2

3 1,

2 2

2 2,

2 2

1 3,

2 2

3 1,

2 2

2 2,

2 2

1 3,

2 2

1 3,

2 2

2 2,

2 2

3 1,

2 2

30 45

60

90 120

135

150

180

210

225 240

315

270

300 330

(-1, 0)

(0, -1)

(1, 0)

(0, 1)

0°

Example

Find the following values using the unit circle.

a. cos 225°

b. sin 300°

c. sin 150°

d. cos 60°

2cos225 0.707

2

3sin300 0.866

2

1sin150 0.5

2

1cos60 0.5

2

Example

Find the exact values of cos (-120°) and

sin (-120°).

1. Make a sketch of a -120° angle.

-270°

-180°

-90°

x

y

0° -360°

-120°

2. Find the positive coterminal angle of -120°.

-120° + 360° = 240°

3. Use the unit circle to find the cosine and sine

of this coterminal angle.

4. These are the answers for the negative angle

measure.

1cos240

2

3sin 240

2

1

cos 1202

3

sin 1202