Functions Modeling Change A Preparation for Calculus Third Edition

Transcription

1 Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1

2 CHAPTER 6 TRIGNMETRIC FUNCTINS SECTIN 6.3 RADIANS

3 So far we have measured angles in degrees. There is another way to measure an angle, which involves arclength. This is the idea behind radians; it turns out to be very helpful in calculus. Page 257 3

4 Definition of a Radian The arc length spanned, or cut off, by an angle is shown below. Page 257 4

5 An angle of 1 radian is defined to be the angle, in the counterclockwise direction, at the center of a unit circle which spans an arc of length 1. Page 257 Blue Box 5

6 In other words, 1 radian cuts off an arc length of 1 in a unit circle. Page 257 6

7 An angle of 2 radians cuts off an arc length of 2 in a unit circle. Page 258 7

8 An angle of 0.6 radian is measured clockwise and cuts off an arc of length 0.6. Page 258 8

9 In general: The radian measure of a positive angle is the length of the arc spanned by the angle in a unit circle. For a negative angle, the radian measure is the negative of the arc length. Page 258 Blue Box 9

10 Radians are dimensionless units of measurement for angles (they do not have units of length). Page

11 Relationship Between Radians and Degrees Page

12 The circumference, C, of a circle of radius r is given by: C 2 r Page

13 What if the circle has a radius of 1? C 2 r Page

14 C 2 Page

15 This means that the arc length spanned by a complete revolution of 360 is 2π, so radians Page

16 Dividing by 2π gives: radian = radian = Page

17 ne-quarter revolution, or 90, is equal to: 1 (2 )= radians 4 2 Page

18 rad. rad rad Page N/A 18

19 rad. rad rad rad. rad rad. 2 2 Page N/A 19

20 rad. rad rad rad. rad rad rad. rad rad Page N/A 20

21 rad. rad rad rad. rad rad rad. rad rad rad. 2 rad rad. 1 1 Page N/A 21

22 Page

23 In which quadrant is an angle of 2 radians? An angle of 5 radians? Page 258 Example #1 23

24 Page

25 360 1 radian = Page N/A 25

26 360 1 radian = radians = Page N/A 26

27 360 1 radian = radians = radians = Page N/A 27

28 360 1 radian = radians = radians = radians = Page N/A 28

29 Page N/A radian = radians = radians = radians = radians =

30 The second quadrant includes angles between π/2 and π, (between and radians), so 2 radians lies in the second quadrant. Page 258 Example #1 30

31 An angle of 5 radians is between 4.71 and radians, that is, between 3π/2 and 2π radians, so 5 radians lies in the fourth quadrant. Page 258 Example #1 31

32 The ferris wheel described in Section 6.1 makes one rotation every 30 minutes. (a) If you start at the 3 c'clock position, find the angle in radians that specifies your position after 10 minutes. (b) Find the angle that specifies your position after t minutes. Page 258 Example #2 32

33 The ferris wheel described in Section 6.1 makes one rotation every 30 minutes. (a) If you start at the 3 c'clock position, find the angle in radians that specifies your position after 10 minutes. Page 258 Example #2 33

34 The ferris wheel described in Section 6.1 makes one rotation every 30 minutes. (a) If you start at the 3 c'clock position, find the angle in radians that specifies your position after 10 minutes. 10 minutes 1 of a rotation 30 minutes 3 Page 258 Example #2 34

35 The ferris wheel described in Section 6.1 makes one rotation every 30 minutes. (a) If you start at the 3 c'clock position, find the angle in radians that specifies your position after 10 minutes. 10 minutes 1 of a rotation 30 minutes Angle= 2 radians 3 3 Page 258 Example #2 35

36 The ferris wheel described in Section 6.1 makes one rotation every 30 minutes. (b) Find the angle that specifies your position after t minutes. Page 258 Example #2 36

37 The ferris wheel described in Section 6.1 makes one rotation every 30 minutes. (b) Find the angle that specifies your position after t minutes. t minutes 30 minutes 30 t of a rotation Page 258 Example #2 37

38 Converting Between Degrees and Radians Page

39 Converting Between Degrees and Radians To convert degrees to radians, or vice versa, we use the fact that 2π radians = 360. Page

40 Converting Between Degrees and Radians To convert degrees to radians, or vice versa, we use the fact that 2π radians = radian = Page

41 Converting Between Degrees and Radians To convert degrees to radians, or vice versa, we use the fact that 2π radians = radian = Similarly, Page radians

42 Converting Between Degrees and Radians a) Convert 3 radians to degrees. b) Convert 3 degrees to radians radian = Page 259 Example # radians

43 Converting Between Degrees and Radians a) Convert 3 radians to degrees. b) Convert 3 degrees to radians radians radians radians Page 259 Example #3 43

44 Converting Between Degrees and Radians a) Convert 3 radians to degrees. b) Convert 3 degrees to radians radians radians radians radians radians (3 ) rad Page 259 Example #3 44

45 The word radians is often dropped, so if an angle or rotation is referred to without units, it is understood to be in radians. We can write, for instance, 90 = π/2 and π = radians radians radians radians radians (3 ) rad Page 259 Example #3 45

46 Arc Length We defined a radian using arc length in a unit circle. An angle of θ radians spans an arc of length θ in a unit circle. However, radians can be used to calculate arc length in a circle of any size. An angle of θ radians spans an arc of length rθ in a circle of radius r. See the following graphic: Page

47 Arc Length We defined a radian using arc length in a unit circle. An angle of θ radians spans an arc of length θ in a unit circle. Page

48 Arc Length The arc length, s, spanned in a circle of radius r by an angle of θ radians, 0 θ 2π, is given by: s = rθ Page 260 Blue Box 48

49 Thus if the size of an angle is fixed, the arc length it spans is proportional to the radius of the circle. Note that θ must be in radians in this arc length formula. Page

50 n a circle of radius 1, one radian spans an arc of length 1. Page

51 n a circle of radius 2, one radian spans an arc of length 2. Page

52 n a circle of radius 3, one radian spans an arc of length 3. Page

53 What length of arc is cut off by an angle of 120 on a circle of radius 12 cm? Page 260 Example #5 53

54 Convert 120 to radians: Page 260 Example #5 54

55 Angle = 120 radians Page 260 Example #5 55

56 Arc Length = (12) cm. Page 260 Example #5 56

57 TI: Mode = Radian radians Page 260 Example #5 57

58 S=r (12 cm.)( ) cm. Page 260 Example #5 58

59 Put in terms of : cm. = cm. Page 260 Example #5 59

60 End of Section