1. Objectives
Convert angle measures between degrees and radians.

2. So far, you have measured angles in degrees
So far, you have measured angles in degrees. You can also measure angles in radians.
A radian is a unit of angle measure based on arc length. Recall from geometry that an arc is an unbroken part of a circle. If a central angle θ in a circle of radius r, then the measure of θ is defined as 1 radian.

3. The circumference of a circle of radius r is 2r
The circumference of a circle of radius r is 2r. Therefore, an angle representing one complete clockwise rotation measures 2 radians. You can use the fact that 2 radians is equivalent to 360° to convert between radians and degrees.

5. Example 1: Converting Between Degrees and Radians
Convert each measure from degrees to radians or from radians to degrees.
A. – 60° . B.

6. Example 2
Convert each measure from degrees to radians or from radians to degrees.
a. 80° 4 9 . b. 20 .

8. Example 3: Automobile Application
A tire of a car makes 653 complete rotations in 1 min. The diameter of the tire is 0.65 m. To the nearest meter, how far does the car travel in 1 s?
Step 1 Find the radius of the tire.
The radius is of the diameter.
Step 2 Find the angle θ through which the tire rotates in 1 second.
Write a proportion.

9. Example 3 Continued
The tire rotates θ radians in 1 s and 653(2) radians in 60 s.
Cross multiply.
Divide both sides by 60.
Simplify.

10. Example 3 Continued
Step 3 Find the length of the arc intercepted by
radians.
Use the arc length formula.
Substitute for r and for θ
Simplify by using a calculator.
The car travels about 22 meters in second.

11. Example 4
A minute hand on Big Ben’s Clock Tower in London is 14 ft long. To the nearest tenth of a foot, how far does the tip of the minute hand travel in 1 minute?
Step 1 Find the radius of the clock.
r =14
The radius is the actual length of the hour hand.
Step 2 Find the angle θ through which the hour hand rotates in 1 minute.
Write a proportion.

12. Example 4 Continued
The hand rotates θ radians in 1 m and 2 radians in 60 m.
Cross multiply.
Divide both sides by 60.
Simplify.

13. Example 4 Continued
Step 3 Find the length of the arc intercepted by
radians.
Use the arc length formula.
Substitute 14 for r and for θ.
s ≈ 1.5 feet
Simplify by using a calculator.
The minute hand travels about 1.5 feet in one minute.