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