1. Trigonometric Functions of Angles 6

2. Chapter Overview The trigonometric functions can be defined in two different but equivalent ways: As functions of real numbers (Chapter 5) As functions of angles (Chapter 6)

3. Chapter Overview The two approaches to trigonometry are independent of each other. So, either Chapter 5 or Chapter 6 may be studied first. We study both approaches because different applications require that we view these functions differently.

4. Chapter Overview The approach in this chapter lends itself to: Geometric problems involving finding angles and distances

5. Chapter Overview Suppose we want to find the distance to the sun. Using a tape measure is, of course, impractical. So, we need something besides simple measurement to tackle this problem.

6. Chapter Overview Angles are easy to measure. For example, we can find the angle formed by the sun, earth, and moon by simply pointing to the sun with one arm and the moon with the other and estimating the angle between them.

7. Chapter Overview The key idea then is to find a relationship between angles and distances. So, if we had a way to determine distances from angles, we’d be able to find the distance to the sun without going there. The trigonometric functions provide us with just the tools we need.

8. Chapter Overview If ABC is a right triangle with acute angle θ as shown, then we define sin θ to be the ratio y/r.

9. Chapter Overview Triangle A'B'C is similar to triangle ABC. Therefore,

10. Chapter Overview Although the distances y' and r' are different from y and r, the given ratio is the same.

11. Chapter Overview Thus, in any right triangle with acute angle θ, the ratio of the side opposite angle θ to the hypotenuse is the same and is called sin θ. The other trigonometric ratios are defined similarly.

12. Chapter Overview In this chapter, we learn how trigonometric functions can be used to measure distances on the earth and in space. In Exercises 61 and 62, we actually determine the distance to the sun using trigonometry.

13. Chapter Overview Right triangle trigonometry has many other applications. Determining the optimal cell structure in a beehive (Exercise 67) Explaining the shape of a rainbow (Exercise 69) Mapping a town (Focus on Modeling)

14. Angle Measure 6.1

15. Angle An angle AOB consists of two rays R 1 and R 2 with a common vertex O. We often interpret an angle as a rotation of R 1 onto R 2.

16. Initial and Terminal Sides In this case, R 1 is called the initial side of the angle. R 2 is called the terminal side of the angle.

17. Positive and Negative Angles If the rotation is counterclockwise, the angle is considered positive. If it is clockwise, the angle is considered negative.

18. Angle Measure

19. The measure of an angle is the amount of rotation about the vertex required to move R 1 onto R 2. Intuitively, this is how much the angle “opens.”

20. Degree One unit of measurement for angles is the degree. An angle of measure 1 degree is formed by rotating the initial side of a complete revolution.

21. Radian Measure In calculus and other branches of mathematics, a more natural method of measuring angles is used—radian measure. The amount an angle opens is measured along the arc of a circle of radius 1 with its center at the vertex of the angle.

22. Radian Measure—Definition If a circle of radius 1 is drawn with the vertex of an angle at its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle.

23. Radian Measures The circumference of the circle of radius 1 is 2π. So, a complete revolution has measure 2π rad.

24. Radian Measures A straight angle has measure π rad. A right angle has measure π/2 rad. An angle that is subtended by an arc of length 2 along the unit circle has radian measure 2.

25. Relationship Between Degrees and Radians A complete revolution measured in degrees is 360° and measured in radians is 2π rad. So, we get the following simple relationship between these methods of angle measurement.

26. Relationship Between Degrees and Radians To convert degrees to radians, multiply by. To convert radians to degrees, multiply by.

27. Relationship Between Degrees and Radians To get some idea of the size of a radian, notice that: 1 rad ≈ 57.296° 1° ≈ 0.01745 rad An angle θ of measure 1 rad is shown here.

28. E.g. 1—Converting Between Radians and Degrees (a)Express 60° in radians. (b)Express π/6 rad in degrees.

29. Terminology Here are a few points to note about terminology. We often use a phrase such as “a 30° angle” to mean an angle whose measure is 30°. For an angle θ, we write θ = 30° or θ = π/6 to mean the measure of θ is 30° or π/6 rad. When no unit is given, the angle is assumed to be measured in radians.

30. Angles in Standard Position

31. Standard Position An angle is in standard position if it is drawn in the xy-plane with its: Vertex at the origin. Initial side on the positive x-axis.

32. Coterminal Angles Two angles in standard position are coterminal if their sides coincide. Here, the angles in (a) and (c) are coterminal.

33. E.g. 2—Coterminal Angles (a)Find angles that are coterminal with the angle θ = 30° in standard position. (b)Find angles that are coterminal with the angle θ = π/3 in standard position.

34. E.g. 2—Coterminal Angles To find positive angles that are coterminal with θ, we add any multiple of 360°. Thus, 30° + 360° = 390° and 30° + 720° = 750° are coterminal with θ = 30°. Example (a)

35. E.g. 2—Coterminal Angles To find negative angles that are coterminal with θ, we subtract any multiple of 360°. Thus, 30° – 360° = –330° and 30° – 720° = –690° are coterminal with θ. Example (a)

36. E.g. 2—Coterminal Angles To find positive angles that are coterminal with θ, we add any multiple of 2π. Thus, are coterminal with θ = π/3. Example (b)

37. E.g. 2—Coterminal Angles To find negative angles that are coterminal with θ, we subtract any multiple of 2π. Thus, are coterminal with θ. Example (b)

38. E.g. 3—Coterminal Angles Find an angle with measure between 0° and 360° that is coterminal with the angle of measure 1290° in standard position.

39. E.g. 3—Coterminal Angles We can subtract 360° as many times as we wish from 1290°. The resulting angle will be coterminal with 1290°. So, 1290 ° – 360 ° = 930 ° is coterminal with 1290 °. So is the angle 1290 ° – 2(360) ° = 570 °.

40. E.g. 3—Coterminal Angles To find the angle we want between 0° and 360°, we subtract 360° from 1290° as many times as necessary. An efficient way to do this is to determine how many times 360° goes into 1290°. That is, divide 1290 by 360, and the remainder will be the angle we are looking for.

41. E.g. 3—Coterminal Angles We see that 360 goes into 1290 three times with a remainder of 210. Thus, 210° is the desired angle.

42. Length of a Circular Arc

43. An angle whose radian measure is θ is subtended by an arc that is the fraction θ/(2π) of the circumference of a circle.

44. Length of a Circular Arc Thus, in a circle of radius r, the length s of an arc that subtends the angle θ is:

45. Length of a Circular Arc In a circle of radius r, the length s of an arc that subtends a central angle of θ radians is: s = r θ

46. Length of a Circular Arc Solving for θ, we get the important formula This formula allows us to define radian measure using a circle of any radius r, as follows.

47. Length of a Circular Arc The radian measure of an angle θ is s/r, where s is the length of the circular arc that subtends θ in a circle of radius r.

48. E.g. 4—Arc Length and Angle Measure (a)Find the length of an arc of a circle with radius 10 m that subtends a central angle of 30°. (b)A central angle θ in a circle of radius 4 m is subtended by an arc of length 6 m. Find the measure of θ in radians.

49. E.g. 4—Arc Length From Example 1 (b), we see that: 30° = π/6 rad So, the length of the arc is: Example (a)

50. E.g. 4—Angle Measure By the formula θ = s/r, we have: Example (b)

51. Area of a Circular Sector

52. The area of a circle of radius r is A = πr 2. A sector of this circle with central angle θ has an area that is the fraction θ/2π of the area of the entire circle.

53. Area of a Circular Sector So, the area of this sector is:

54. Area of a Circular Sector In a circle of radius r, the area A of a sector with a central angle of θ radians is:

55. E.g. 5—Area of a Sector Find the area of a sector of a circle with central angle 60° if the radius of the circle is 3 m. To use the formula for the area of a circular sector, we must find the central angle of the sector in radians: 60° = 60(π/180) rad = π/3 rad

56. E.g. 5—Area of a Sector Thus, the area of the sector is:

57. Circular Motion

58. Suppose a point moves along a circle as shown. There are two ways to describe the motion of the point: Linear speed Angular speed

59. Linear Speed Linear speed is the rate at which the distance traveled is changing. So, linear speed is the distance traveled divided by the time elapsed.

60. Angular Speed Angular speed is the rate at which the central angle θ is changing. So, angular speed is the number of radians this angle changes divided by the time elapsed.

61. Linear Speed and Angular Speed Suppose: A point moves along a circle of radius r. The ray from the center of the circle to the point traverses θ radians in time t.

62. Linear Speed and Angular Speed Let s = rθ be the distance the point travels in time t. Then, the speed of the object is given by:

63. E.g. 6—Finding Linear and Angular Speeds A boy rotates a stone in a 3-ft-long sling at the rate of 15 revolutions every 10 seconds. Find the angular and linear velocities of the stone.

64. E.g. 6—Finding Angular Speed In 10 s, the angle θ changes by 15 · 2π = 30π radians So, the angular speed of the stone is:

65. E.g. 6—Finding Linear Speed The distance traveled by the stone in 10 s is: s = 15 · 2πr = 15 · 2π · 3 = 90π ft So, the linear speed of the stone is:

66. Linear Speed and Angular Speed Notice that angular speed does not depend on the radius of the circle, but only on the angle θ. However, if we know the angular speed ω and the radius r, we can find linear speed as follows: v = s/t = rθ/t = r(θ/t) = rω

67. Relationship Between Linear and Angular Speeds If a point moves along a circle of radius r with angular speed ω, then its linear speed v is given by: v = r ω

68. E.g. 7—Finding Linear Speed from Angular Speed A woman is riding a bicycle whose wheels are 26 inches in diameter. If the wheels rotate at 125 revolutions per minute (rpm), find the speed at which she is traveling, in mi/h.

69. The angular speed of the wheels is: 2π · 125 = 250π rad/min. Since the wheels have radius 13 in. (half the diameter), the linear speed is: v = rω = 13 · 250π ≈ 10,210.2 in./min E.g. 7—Finding Linear Speed from Angular Speed

70. There are 12 inches per foot, 5280 feet per mile, and 60 minutes per hour. So, her speed in miles per hour is: E.g. 7—Finding Linear Speed from Angular Speed