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Notes 13.1
Angles and the Unit Circle
Copyright © 2005 Pearson Education, Inc.
Standard Position Angles

Standard Position if its vertex
An angle is in _________________
is at the origin and its initial side is along the
positive x-axis.

Angles in standard position having their terminal
sides along the x-axis or y-axis, such as angles
with measures 90, 180, 270, and so on, are
called _______________________________.
Quadrantal Angles
Copyright © 2005 Pearson Education, Inc.
Slide 1-2
STANDARD POSITION Angle

Angle - formed by rotating
a ray around its endpoint.

The ray in its initial position
is called the
____________________
Initial side
of the angle.
The ray in its location after
the rotation is the
____________________
Terminal side
of the angle.

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Slide 1-3

Positive Angle
____________________
The rotation of the
terminal side of an angle
counterclockwise.
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
Negative Angle
____________________
The rotation of the
terminal side is clockwise.
Slide 1-4

A complete rotation of a ray results in an angle
measuring 360. By continuing the rotation, angles of
measure larger than 360 can be produced. Such angles
are called ____________________________
Coterminal Angles
Copyright © 2005 Pearson Education, Inc.
Slide 1-5
Example: Coterminal Angles


Find the angle of smallest possible positive measure
coterminal with each angle.
a) 1115
b) 187
Add or subtract 360 as many times as needed to obtain an
angle with measure greater than 0 but less than 360.
A. 1115  (360)  755
B. 187 + 360 = 173
A. 1115  2(360)  395
A. 1115  3 (360)  35
Copyright © 2005 Pearson Education, Inc.
Slide 1-6
The unit circle can be broken into degrees or radians
Why radians?
Radian measure is important to mathematics, especially trigonometry and
calculus. It allows for very simple expression of derivative and integral
relations that involve trigonometric functions in calculus.
Definition of one Radian
r
S=r
Θ
r
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A radian is a ratio of the intercepted arc to
the central angle so ONE radian is the
measure when the intercepted arc “S” is
equal to the radius of the circle
Slide 1-7
The unit circle can be broken into degrees or radians
How many degrees are in a circle?
360°
If the unit circle has a radius of 1, what is it’s circumference?
π
If 360° corresponds to 2π, 180° corresponds to ____
2π

2
90° ? ___
Conversions Between Degrees and Radians
1. To convert degrees to radians, multiply degrees by
2. To convert radians to degrees, multiply radians by
Copyright © 2005 Pearson Education, Inc.
8
3
2
270°? ____

180
180

Slide 1-8
Ex 1. Convert the degrees to radian measure.
60
- 54
.
.

180

180
Copyright © 2005 Pearson Education, Inc.



3
radians
 3
10
radians
Slide 1-9
Ex 2. Convert the radians to degrees.
a)

6
b)
11

18
.
180

.
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180

 30
  110
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Ex 3. Find one positive and one negative angle that is

coterminal with the angle  =
in standard position.
3
POSITIVE ANGLE
NEGATIVE ANGLE

 2
3

 2
3
 6

3
3
 6

3
3
7
3
 5
3
Copyright © 2005 Pearson Education, Inc.
Slide 1-11
90
Complementary angles: Two angles with a sum of ______________
180
Supplementary angles: Two angles with a sum of ______________
Find the complement of angle θ = 36°
90  36
 54
Find the supplement of angle θ = 12°
180  12
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 168
Slide 1-12

2
Complementary angles: Two angles with a sum of ______________

Supplementary angles: Two angles with a sum of ______________

Find the complement of angle θ =
4
 

2 4
2 

4
4


4
2
Find the supplement of angle θ =
15
2

15
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15 2

15
15
13

15
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