1. Chapter 4 4-1 Radian and degree measurement

2. Objectives O Describe Angles O Use radian measure O Use degree measure and convert between and radian measure

3. Angles O As derived form the Greek language O Trigonometry means “measurement of triangles “ O Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation and surveying O Today, the use has expanded to involve rotations, orbits, waves, vibrations, etc.

4. Angles O An angle is determined by rotating a ray (half-line) about its endpoint.

5. Definitions O The initial side of an angle is the starting position of the rotated ray in the formation of an angle. O The terminal side of an angle is the position of the ray after the rotation when an angle is formed. O The vertex of an angle is the endpoint of the ray used in the formation of an angle.

6. Standard Position O An angle is in standard position when the angle’s vertex is at the origin of a coordinate system and its initial side coincides with the positive x-axis.

7. Positive and negative angles O A positive angle is generated by a counterclockwise rotation; whereas a negative angle is generated by a clockwise rotation.

8. Coterminal O If two angles are coterminal, then they have the same initial side and the same terminal side.

9. Radian Measure O The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. O One way to measure angles is in radians. O To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle.

10. Radian Measure

11. Radian

12. How many radians are in a circle?

13. Radians O This means that the circle itself contains an angle of rotation of 2 π radians. Since 2 π is approximately 6.28, this matches what we found above. There are a little more than 6 radians in a circle. (2 π to be exact.) Therefore: A circle contains 2 π radians. A semi-circle contains π radians of rotation. A quarter of a circle (which is a right angle) contains 2 radians of rotation

15. Radians

16. Coterminal angles

17. Example#1

18. Example#2

19. Example#3

20. Student guided practice O Do problems 25 and 26 in your book page 261

21. Degree Measure

26. Example O Example: Convert 120° to radians. O Example: Convert -315° to radians

27. Example

28. Student guided practice O Do odd problems form 55-65 in your book page 262

30. Acute and Obtuse

31. Example

32. Arc Length O Because we already know that with radian measure θ =r /s, O where s is the arc length, then s = r θ.

33. Example O Example: Find the length of the arc that subtends a central angle with measure 120° in a circle with radius 5 inches.

34. Example O A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240 degrees, as shown in Figure 4.15.

35. Student Guided practice O Do problem 93 and 94 in your book page 263

36. Linear and angular speed

38. Example O Example: The circular blade on a saw rotates at 2400 revolutions per minute. O (a) Find the angular speed in radians per second. O (b) The blade has a diameter of 16 inches. Find the linear speed of a blade tip.

39. Area of a sector

40. Homework O Do problems 27,28,45,46,51,52,56,58,79,85 O In your book page 261 and 262

41. Closure O Today we learned about radian and degree measure O Next class we are going to learn about the unit circle