N
O
LE
SS
18
The Unit Circle
UNDERSTAND Angles and arcs can be measured in radians as well as in degrees. A circle
contains 360° and 2p radians. So, 180° is equal to p radians.
p radians
• To convert from degrees to radians, multiply by ​ _______
 ​. 
 
180°
180°
   ​. 
• To convert from radians to degrees, multiply by _______
​ p radians
UNDERSTAND A unit circle has its center at the origin and a radius of 1 unit.
y
y
(0, 1)
2
(1, 0)
2
3 3
120° 90°
5 4
135°
6
150°
180°
(0, 1)
3 4 60°
45°
6
30°
0°
210°
330°
11
7
225°
315°
6
6 5
240°
270° 300° 7
4
5 4
4
3
3
3
2
(0, 1)
Terminal
Side
0
(1, 0)
40°
(1, 0)
x
Initial Side
(1, 0)
x
(0, 1)
The measure of an angle in radians is related to the length of the arc that it cuts in the unit
circle. In a circle with radius r, a central angle created by an arc of length r measures 1 radian.
y
UNDERSTAND Positive angles are measured in a
counterclockwise direction, starting from 0°, as shown by
the 40° angle above. A negative angle measure indicates
moving from 0° in a clockwise direction.
(1, 0)
For angle measures greater than 360° or 2p radians, the ray
has gone around the circle completely and then started over
again from 0°. That is, an angle measuring 380° would go
around the circle once and then another 20° past the positive
x-axis. So, a 380° angle and a 20° angle look the same on
the unit circle. Angles with different measurements whose
terminal sides lie on the same ray are coterminal. Note that a
2340° angle would also be coterminal with 380° and 20° angles.
166 (0, 1)
340°
20°
380°
(1, 0)
(0, 1)
x
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An angle is a rotation of a ray around its endpoint. The angle shown above in the graph on the
right is a rotation from 0° to 40°. The ray through 0° is the initial side of the angle. The ray
through 40° is the terminal side of the angle. This is the standard position for an angle: Its
vertex is at the origin, and its initial side is on the positive x-axis.
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Connect
5p
Draw an angle that measures ___
​  4   ​ radians in standard position on the unit circle.
1
Determine in which quadrant the angle’s
terminal side will be. Use Math Tool: Unit
Circle to help you.
5p
___
​  4   ​ is between p and 2p.
So, the angle’s terminal side will be on the
bottom half of the circle.
5p
3p
1
1
___
___
__
​  4   ,​  or 1​ __
4  ​ p, is between p and ​  2   ,​  or 1​  2 ​  p.
So, the angle’s terminal side will be in
Quadrant III.
2
Determine where in Quadrant III the
terminal side will be.
A rotation of p radians goes through half of
a full rotation, or to the negative x-axis.
Subtract p from the total angle measure:
3
5p
Draw the angle.
p
__
​ ___
4   ​ 2 p 5 ​  4 ​
p
A measure of __
​ 4 ​radians means one-fourth
From the negative x-axis, rotate another
From the negative x-axis, rotate halfway
direction.
of a half circle or one-eighth of a full circle.
p
__
​  4 ​ radians in the positive (counterclockwise)
through Quadrant III.
5p
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​  4   ​ radians is
▸ An angle that measures ___
shown here.
y
(0, 1)
(1, 0)
(1, 0)
5
4
(0, 1)
x
SC U S S
DI
5p
What is ___
​  4   ​ radians in degrees? Is it
easier for you to draw the angle with the
measure in degrees or in radians?
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EXAMPLE A Find and draw one positive angle and one negative angle that are coterminal with an
2p
angle that measures ___
​  3   ​ radians.
1
Draw an angle with a measure of 2p
___
​  3   ​ radians.
2p
2
___
​  3   ​ radians, or __
​ 3 ​ p radians, is two-thirds of
the distance from 0 to p on a unit circle.
y
2
3
(0, 1)
(1, 0)
(1, 0)
x
2
Find a positive angle that is coterminal.
Add a full rotation to the angle.
(0, 1)
A full rotation is 2p radians.
2p
6p
2p
___ ___
​ ___
3   ​ 1 2p 5 ​  3   ​ 1 ​  3   ​ 
8p
5 ___
​  3   ​ 
3
Find a negative angle that is coterminal.
2p
2p
6p
___ ___
​ ___
3   ​ 2 2p 5 ​  3   ​ 2 ​  3   ​ 
4
Draw the coterminal angles.
4p
5 2 ​ ___
3   ​ 
2p 8p
4p
​  3   , ​ ___
​  3   ,​  and 2 ​ ___
▸ Angles that measure ___
3   ,​ 
y
2
3
(0, 1)
in radians, are coterminal.
4
3
8
3
(1, 0)
(1, 0)
TRY
Find one positive angle and one negative
angle that are coterminal with 50°.
168 (0, 1)
x
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Subtract a full rotation from the angle.
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EXAMPLE B A ray is rotated to create an angle, as shown on the unit circle. Find the measure of the
angle in radians. Then, convert the measure to degrees.
y
(0, 1)
(1, 0)
(1, 0)
x
(0, 1)
1
Examine the rotation.
The arrow shows that the direction of the
rotation is clockwise, or in the negative
direction. So, you know the angle measure
will be negative.
The rotation goes around the circle one full
time, then continues for a portion of the
circle. This tells you that the absolute value
of the angle’s measure in radians will be
greater than that of a full rotation (greater
than 2p).
2
Find the angle measure in radians.
The rotation is one full circle plus one
quarter of a circle.
A full circle is 2p radians.
2p
p
​  4   ,​  or __
A quarter circle is ___
​  2 ,​ radians.
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Both rotations are in the negative direction,
so both measures are negative.
Add the rotations:
Convert from radians to degrees.
180°
Multiply by _______
​ p radians
   ​. 
5p
5p
___
22p 1 (​ 2 ​ __
2 ​  )​ 5 2 ​  2 ​  
p
3
5p
The angle measures 2 ​ ___
2 ​  radians.
180°
_______
   ​ 
5 2450°
2 ​ ___
2 ​  radians ? ​  p radians
5p
▸ The angle shown measures 2 ​ ___
2 ​  radians,
or 2450°.
CHECK
Find a positive angle measure between
0 and 2p radians (or 0° and 360°) that is
coterminal with the solution above.
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Practice
Write an appropriate word or phrase in each blank.
1.
A circle is made up of 360° or 2p
has its center at the origin and a radius of 1 unit.
2.A(n)
3.
.
An angle in standard position has its
side on the positive x-axis and its
side at the end of the rotation.
4.
Angles with different measurements that lie on the same ray are
.
Convert the measure to the indicated units.
3p
5.​ ___
4   ​ radians 5
°
radians
6.60° 5
radians
7.240° 5
8.
7p
2 ​ ___
8 ​  radians 5
REMEMBER A measure will be
positive or negative in both forms.
°
Draw the indicated angle.
4p
p
9.​ ___
3   ​ radians
10. 2 ​ __
4 ​ radians
y
y
(0, 1)
(1, 0)
(1, 0)
x
(1, 0)
(0, 1)
(1, 0)
x
(0, 1)
REMEMBER A negative measure is
a clockwise rotation from 0 radians.
For each angle measure, find one positive and one negative angle, in radians, that is coterminal.
p
11.​ __
9 ​ radians p
12. 2 ​ __
 ​ radians 6
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(0, 1)
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Approximate the angle measure shown, in radians.
y
13.
y
14.
(0, 1)
(0, 1)
(1, 0)
(1, 0)
(0, 1)
x
(1, 0)
(1, 0)
x
(0, 1)
Solve.
15.
CONNECT In geography, latitude gives the north-south position of a point on Earth’s surface.
Latitude is an angle, ranging from 0° at the equator to 90°N (or 90°North) at the North Pole and
90°S (or 90°South) at the South Pole. The overland distance between points on Earth’s surface is
an arc, with the central angle measured from Earth’s center.
The formula for the length, L, of an arc with central angle x° on a circle with radius r is:
x
L 5 2pr ? ____
​ 360
   ​ 
Suppose that two cities have the same longitude, which gives the east-west position of a point
on Earth’s surface. One has latitude 23°N, and the other has latitude 47°S. Assume Earth is a
sphere with a radius of 6,380 km. Find the distance between the cities to the nearest kilometer.
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km
Rewrite the arc length formula for an angle measure x given in radians, and simplify.
L5
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