Chapter 4

1. Chapter 4 4-1 Radian and degree measurement

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

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

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

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

6. Standard Position • 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 • A positive angle is generated by a counterclockwise rotation; whereas a negative angle is generated by a clockwise rotation.

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

9. Radian Measure • The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. • One way to measure angles is in radians. • 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 • Arc length = radius when = 1 radian

11. Radian

12. How many radians are in a circle? • *In general, the radian measure of a central angle θ with radius r and arc length s is • θ = s/r. • We know that the circumference of a circle is 2πr. If we consider the arc s as being the circumference, we get θ = 2π r/r =2

13. Radians • This means that the circle itself contains an angle of rotation of 2π radians. Since 2π is approximately 6.28, this matches what we found above. There are a little more than 6 radians in a circle. (2π to be exact.) Therefore: A circle contains 2π radians. A semi-circle contains π radians of rotation. A quarter of a circle (which is a right angle) contains 𝜋2 radians of rotation

15. Radians

16. Coterminal angles • Two angles are coterminal when they have the same initial and terminal sides. • For instance, the angle 0 and 2 are coterminal, as are the angles . • You can find an angle that is coterminal to a given angle by adding or subtracting 2.

17. Example#1 • For the positive angle 13/ 6, subtract 2to obtain a coterminalangle.

18. Example#2 • For the positive angle 3/ 4, subtract 2to obtain a coterminalangle.

19. Example#3 • For the negative angle –2/ 3, add 2to obtain a coterminalangle.

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

21. Degree Measure • Definition: A degree is a unit of angle measure that is equivalent to the rotation in 1/360th of a circle. • Because there are 360° in a circle, and we now know that there are also 2π radians in a circle, then 2π = 360°. • 360° = 2π radians 2π radians= 360° • 180° = π radians π radians = 180° • 1° = radians 1 radian= 180/

24. Degree Measure • To convert radians to degrees, multiply by . • To convert degrees to radians, multiply by • 180/.

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

27. Example • Example: Convert to degrees. • Convert 7 to degrees.

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

30. Acute and Obtuse • An acute angle has a measure between • 0 and • (or between 0° and 90°.) • An obtuse angle has a measure between and π (or between 90° and 180°.)

31. Example • Example: Find the supplement and complement of

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

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

34. Example • 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 • Do problem 93 and 94 in your book page 263

36. Linear and angular speed • Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed of the particle is • Linear speed = arc length/ time= s/t =

37. Linear and angular speed • Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is Angular speed = central angle/ time= t.

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

39. Area of a sector

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

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