1. 4-1 : Angles and Their Measures What you’ll learn about ■ The Problem of Angular Measure ■ Degrees and Radians ■ Circular Arc Length ■ Angular and Linear Motion... and why Angles are the domain elements of the trigonometric functions.

2. Getting Started Darla watches Larry walk around a circle. The circle’s radius is 1 meter. Darla stands at the center and Ben begins walking counterclockwise. Consider a coordinate grid, with Darla standing at the origin and Ben starting at point (1,0).

3. Consider Ben’s location after certain distances How far will Ben walk before returning to the point (1,0)?

4. Consider Ben’s location after certain distances Where will Ben be after walking exactly π meters?

5. Consider Ben’s location after certain distances Where will Ben be after walking exactly π/2 meters?

6. Consider Ben’s location after certain distances Where will Ben be after walking exactly 3π meters?

7. Consider Ben’s location after certain distances Where will Ben be after walking exactly 3 meters?

8. Radian Measure

10. Where is Ben after he has walked exactly π/4 meters? How about after 3π/4? π/4 3π/4 2π/4 = π/2

11. At some point Ben has walked exactly 9π/4 meters. Find 2 other distances Ben could have walked around the circle to end up there.

12. How far will Ben walk when he reaches the point (0, -1)?

13. As Ben continues to walk, he will reach (0, -1) again. Give another distance that Ben could walk to reach (0, -1).

14. As Ben continues to walk, he will reach (0, -1) again. Describe a method you could use to generate a large number of these distances.

15. Ben runs 100 meters along the circle. What quadrant is he in at the end of this 100-meter run?

17. Example 1

18. Example 2

20. Example Converting from Degrees to Radians

21. Example Converting from Radians to Degrees If you have a calculator with a “radian-to-degree” conversion key, try using it to verify the result shown in part c

22. On your own: 1.What quadrant will Austin be in if he walks around the unit circle for 45 radians? 2. Convert from degrees to radians: a)30°b) 350° c) -52° 3. Convert from radians to degrees: a) π/5b) 5π/7 c) -33 1 st Quadrant π/6 35π/18 -13π/45 36°≈ 128.6° -5940/π °

23. Arc Length

24. Example

25. On your own: Find the perimeter of a 45° slice of a large (10 in radius) pizza. s = 7.9 P =10 + 10 + s P = 27.9 in

26. v = wr = s/t s = θr w = θ/t

27. Variables to know: θ = angle (in radians) r = radius (distance) s = arc length (distance) t = time v = linear velocity (distance/time) ω = angular velocity (radians/time)

28. Q: The second hand of a clock is 10.2 centimeters long, as shown in Figure. Find the linear speed of the tip of this second hand as it passes around the clock face.

29. Q: A Ferris wheel with a 50-foot radius makes 1.5 revolutions per minute. a. Find the angular speed of the Ferris wheel in radians per minute. b. Find the linear speed of the Ferris wheel.

30. In groups, practice on the worksheet problems