1. 13-3 – Radian Measures

2. Vocabulary and Definitions
A central angle of a circle is an angle with a vertex at the center of the circle.
An intercepted arc is the arc that is “captured” by the central angle.

3. Vocabulary and Definitions
When the central angle intercepts an arc that has the same length as a radius of the circle, the measure of the angle is defined as a radian. r
Like degrees, radians measure the amount of rotation from the initial side to the terminal side of the angle.

4. The Unit Circle 4

5. The “Magic” Proportion
This proportion can be used to convert to and from
Degrees to Radians.
Degrees°
180° =
r radians
radians
Example: Find the radian measure of angle of 45°.
Write a proportion.
45°
180° =
r radians
radians
Write the cross-products.
45 • = 180 • r
Divide each side by 45.
r =
45 •
180
= Simplify. 4
An angle of 45° measures about radians.

6. The “Magic” Proportion
This proportion can be used to convert to and from
Degrees to Radians.
Degrees°
180° =
r radians
radians
Example: Find the radian measure of angle of -270°.
Write a proportion.
-270°
180° =
r radians
radians
Write the cross-products.
-270 • = 180 • r
Divide each side by 45.
r =
-270 •
180
-4.71 Simplify. 2 -3
An angle of -270° measures about radians.

7. Let’s Try Some
Convert the following to radians
a. 390o b. 54o c. 180o

8. Example Find the degree measure of . = Write a proportion.6 13
Write a proportion. 6 13
radians = d°
180
• 180 = • d Write the cross-product. 6 13
d = Divide each side by .
13 • 180
6 • 1 30
= 390° Simplify.
An angle of radians measures 390°. 6 13

9. Example Find the degree measure of an angle of – radians.2 3
– radians • = – radians • 2 3
180°
radians 1 90
Multiply by .
= –270°
An angle of – radians measures –270°. 2 3

10. Radian Measure Find the radian measure of an angle of 54°.
5 4° • radians = 54° • radians Multiply by radians.
180° 3 10 10 3
radians =
Simplify.
An angle of 54° measures radians. 10 3

11. Length of an intercepted arc
For a circle of radius r and a central angle of measure  (in radians), the length s of the intercepted arc is
s = r S  r

12. Radian Measure Use this circle to find length s to the nearest tenth.
s = r Use the formula.
= 6 • Substitute 6 for r and for . 7 6
= 7 Simplify.
22.0 Use a calculator.
The arc has length 22.0 in.

13. Radian Measure
Use this circle to find length s to the nearest tenth.
s = r Use the formula.
= 270 ° Convert to radians. S
= 4• Substitute 4 for r and for . 3 2
270°
4 ft
= 6 Simplify.
18.8 Use a calculator.
The arc has length 18.8 ft.

14. Radian Measure
Another satellite completes one orbit around Earth every 4 h. The satellite orbits 2900 km above Earth’s surface. How far does the satellite travel in 1 h?
Since one complete rotation (orbit) takes 4 h, the satellite completes of a rotation in 1 h. Earth’s radius is 6400 km 1 4
Step 1:  Find the radius of the satellite’s orbit.
r = Add the radius of Earth and the distance from Earth’s surface to the satellite.
= 9300

15. Radian Measure
(continued)
Step 2:  Find the measure of the central angle the satellite travels through in 1 h.
= • 2 Multiply the fraction of the rotation by the number of radians in one complete rotation.
= • Simplify. 1 4 2
Step 3:  Find s for = .
s = r Use the formula.
= 9300 • Substitute 9300 for r and for .
14608 Simplify. 2
The satellite travels about 14,608 km in 1 h.