Measures of Angles and Rotations

1. Section 4-1
Measures of Angles and Rotations

2. Warm-up
A circle has a radius of 1. Find each of the following.
1. Its circumference 2. The length of a 180° arc
π
2π
3. The length of a 45° arc 4. The length of a 210° arc
π 7π
4 6
117.9π
≈ 2.06
5. The length of a 117.9° arc
180

3. Angle: The combination of two sides made up of rays that share
an endpoint, which is the vertex
Measure of an angle: Tells the size and direction of
rotation used to create the angle
A positive rotation is counterclockwise, and a negative rotation
is clockwise
Revolution: Based on what fraction of a circle you would cover
for an angle measure

4. Example 1
Draw a 75° angle. Then find two other angle measures for the angle.
Any angle such that the measure is any integer multiple of a full
revolution greater or smaller

5. Radian: A measure of the arc length of an angle, which gives us
a fraction of a circle
Radian to Degree: In order to switch from radians to
degrees, multiply your radian measure by the following
fractions: 360° 180°
or
π radians
2π radians
Degree to Radian: In order to switch from degrees to
radians, multiply your degree measure by the following
fractions: π radians
2π radians
or
360° 180°

6. Example 2
π
Draw two rays that show a rotation of radians.
4
Any angle such that the measure is any integer multiple of a full
revolution greater or smaller

7. Example 3
Convert 31.4159 radians to degrees
180°
31.4159 • ≈ 1800°
π

8. Example 4
Convert 42° to exact radians and approximate radians
Exact: Approximate:
π 7
π
42° •
180° 30
7
≈ .73 radians
π
=
30

9. Equivalence Table
The values are listed in your textbook on page 234. But how do we
determine these values?

10. Example 5
7
How many revolutions equal π radians?
2
Compare the number of revolutions this would be.
One revolution is 2π , so almost 2 full revolutions.
3
1 revolutions
4

11. Example 6
5
Convert radians to revolutions.
−π
6
5
revolutions
−
12

12. Homework
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