Chapter 3. Radian Measure and Dynamic Trigonometry

Transcription

1 Chapter 3 Radian Measure and Dynamic Trigonometry 1

2 Chapter 3 Topics Angle Measure in Radians Length, Velocity and Area of a Circular sector Unit Circle Trig and Real Numbers 2

3 Chapter 3.1 Angle Measure in Radians 3

4 Overview Radians for Angle Measurement Radian Measure of Standard Angles Converting And yes, we did a bit of this before 4

5 The Unit Circle We use a central circle Circle in the x-y plane Center at the origin A central angle is an angle whose Vertex is at the center of the circle An angle whose sides intersect the circle at B and C subtends an arc BC y C B x 5

6 Radians A radian is the measure of an angle subtended by an arc of whose length is the radius Therefore, the length of an arc, s, the radius r times the angle in radians, 2 r = s Now, lets make some sense of this: What is arc length? The arc whose length is the entire circle is the circumference The circumference is 2 r Therefore, the angle that encompasses the whole circle is 2, which in degree measure is 360 6

7 Example If a circle has radius 8 cm, and the arc length is 18 cm, what is the radian measure of an angle? = s/r = 18/8 = 9/4 = 2.25 rad 7

8 Some More Examples Find the radian measure of the angle with arc s and circle radius r s = 24 m, r = 4 m s = 10 ft, r = 10 ft s = 5.2 mm, r = 2.6 mm s = 11 cm, r = 20 m 8

9 Example (Solutions) Find the radian measure of the angle with arc s and circle radius r s = 24 m, r = 4 m; = 6 s = 10 ft, r = 10 ft; = 1 s = 5.2 mm, r = 2.6 mm; = 2 s = 11 cm, r = 20 m; equivalent to 11 cm and 2000 cm: = 11/2000 9

10 Standard Angles Circumference C = 2 r, or an arc of length 2 r goes around the circle one time This is the same as saying that an arc whose length is the circumference subtends an angle of 2 Therefore, 360 = 2 90 = 2 /4 = /2 180 = 2 /2 = 45 = 2 /8 = /4 60 = 2 /6 = /3 30 = 2 /12 = /6 10

11 Example, Finding Angle Measure Find the radian measure of the following angle: 120 = -225 = 270 = 11

12 Example (Solutions) Find the radian measure of the following angle: 120 = 2(60 ) = 2 ( /3) = 2 /3-225 = - 5(45 ) = -5( /4) = -5 /4 [= 2-5 /4 = 3 /4] 270 = 3(90 ) = 3( /2) 12

13 More Examples Find the following angles in radians 180 = 30 = 330 = -225 = 13

14 More Examples (Solutions) Find the following angles in radians 180 = rad 30 = /6 rad 330 =11 /6 rad -225 = -5 /4 rad 14

15 Converting Degrees and Radians 360 = 2, so 180 = To convert from radians to degrees, multiply by 180 / radians x deg/radians = deg To convert from degrees to radians, multiply by /180 deg x radians/deg = radians 15

16 Example Convert to radians: -75 = = 16

17 Example (Solution) Convert to radians: -75 = -75 ( /180 ) = 5 / = = ( /180 ) =

18 Convert from radians to degrees Example (Solution) / 24 = -5 = 18

19 Convert from radians to degrees Example / 24 = /24 (180 / ) = 180 / 24 = = -5 (180 / ) = 900 / 19

20 More Examples Convert to radians: 230 = -35 = = 3.1, Even problems

21 Convert to radians: 230 = 23 /18 Example (Solutions) -35 =7 / =? For this one, you need to convert 48 to degrees. 48 = 48/60= 0.8 degrees. Then do the multiplication: (200.8/180) 21

22 Convert to Degrees: /4 = /2 = 5 /6 = 6 = 22

23 Solutions Convert to Degrees: /4 = 45 /2 =90 5 /6 =150 6 =

24 Find the terminal side quadrant: = 3.9 More Examples = 5.4 =

25 Find the terminal side quadrant: = 3.9, Q 3 Solutions = 5.4, Q 4 = - 4.3, Q 2 25

26 Summary We can convert from radians to degrees and vice versa: The key is that 360 degrees is 2 radians 26

27 Chapter 3.2 Arc Length, Velocity, Area of Circular Sector 27

28 Overview Use radians to compute the length of a subtended arc Solve problems involving angular velocity and linear velocity Calculate the area of a circular sector 28

29 Length of an arc For, a subtended arc of a circle of radius r, the arc length is s = r, for in radians For example: radius is10 cm, angle is 3.5 radians, subtended arc is s = 10(3.5) = 35 cm 29

30 Angular and Linear Velocity Angular velocity equals the amount of rotation per unit time Often designated as (omega) = / t For example, a Ferris wheel rotating at 10 revolutions per minute has angular velocity of = / t = 10 revolutions/min each revolution is 2 rad = 10 (2 ) / 1 min = 20 rad/min 30

31 Linear Velocity Linear velocity is distance traveled (or change in position) per unit time D = r t, r = D / t For angular motion, the distance D is the length of the arc, s rate = v = s / t Since s = r, v = r / t = r ( / t ) = r Linear velocity v = r 31

32 Example A point P rotates around the circumference of a circle with radius r = 2 ft at a constant rate. If it takes 5 sec to rotate through an angle of 510, a. What is the angular velocity of P b. What is the linear velocity of P 32

33 a. 510 = 510 ( / 180 ) Solution = (510/180) = 17/6 = / t = (17 /6) / 5 = (17 /30) rad/sec b. V = r = 2ft (17 /30 ) = (17 /15) ft/ sec 33

34 More Examples Point P passes through angle in time t as it travels around the circle. Find its angular velocity in radians = 540, t = 9 yr = 270, t = 12 min = 690, t = 5 sec = 300, t = 5 hr 34

35 Solutions Covert to radians, = angle / time = 540, t = 9 yr; 540 = 540 / 180 = 3, = /3 rad/yr = 270, t = 12 min; = /8 rad/min = 690, t = 5 sec; = 23 / 30 rad/sec = 300, t = 5 hr; = /3 rad/hr 35

36 Examples Point P travels around a circle of radius r, find its linear velocity 1. = 12 rad/min, r = 15 ft 2. = 2312 rad/sec, r = 0.01 km 3. = 282, t = 4.1 min, r = 1.2 yd 4. = 45, t = 3 hr, r = 2 mi 36

37 V = r Solutions ft/min mph yd/min mph 37

38 Example Using Angular Velocity to Determine Linear Velocity The wheels on a bicycle have a radius of 13 in. How fast, mph, is the cyclist going if the wheels turn at 300 rpm? = 300 rev/min = 300 (2 ) / min = 600 /min V = r = 13 in (600 /min) 1 mile = 5280 ft x 12 in /ft, 1 hour = 60 min V = 13 in (600 /min) (60 min/hr) ( 1 mi / (5280 x 12 in) V = 23.2 mph 38

39 The Area of a Circular Sector The area of a circle is r 2 The area of one half the circle is r 2 /2 The area subtended by an arbitrary angle = r 2 /2, for in radians 39

40 Examples Find the area of a central angle of 3 /4 if the radius is 72 ft. A = (3 /4 ) (72ft) 2 /2 = 1944 ft 2 40

41 Another Example The second hand on a wristwatch is 12 mm long. Find the area of the watch face the second hand passes over in 20 sec = / t = 2 / 60 rad/sec in 20 sec, = 20 sec (2 / 60 rad/sec) = 2 / 3 rad Area A = r 2 /2 = (2 / 3 rad) (12mm) 2 /2 = 48 mm 2 41

42 Summary Arc Length: s = r, in radians Angular Velocity: = / t, in radians Linear Velocity: v = r Area of a circular sector subtended by angle : r 2 /2, in radians All assume is in radians!!!! 42

43 Chapter 3.3 The Unit Circle 43

44 Locate points on a unit circle Overview Use special triangles to find points on a unit circle Define the 6 trig functions in in terms of points on the unit circle 44

45 Unit Circle Circle of radius 1 Center at the origin (0,1) y (-1,0) (0,0) (1,0) x (0,1) 45

46 Exercise Find a point of the unit circle if y = 1/2 46

47 Solution x 2 + y 2 = 1 x = ± sqrt (1 ¼) = ± sqrt(3)/2 47

48 Symmetry Find the quadrant containing (-3/5, -4/5) and verify it is on the unit circle 48

49 Solution Quadrant : x and y < 0, so is in quadrant 3 Check: x 2 + y 2 = 1 (3/5) 2 + (4/5) 2 = (9 + 16)/25 = 1 49

50 More on Symmetry If the point (a,b) is on the unit circle, so are (-a, b) (a, -b) (-a, -b) 50

51 Special Triangles Find points on the unit circle associated with /4: /4: /2 triangle 51

52 Solution In quadrant 1, the triangle has x and y values sqrt(2)/2 In quadrant 2 it is (- sqrt(2)/2, sqrt(2)/2) In quadrant 3, both signs are In quadrant 4, x is positive, y is negative 52

53 More Examples Do the same for a /6: /3: /4 triangle 53

54 Solutions (± sqrt(3)/2, ± 1/2) (± 1/2, ± sqrt(3)/2) 54

55 Points Associated with Rotation Find the points on the unit circle associated with a. 5 /6 b. 4 /3 c. 7 /4 55

56 Solution a. (-sqrt(3)/2, 1/2) b. (-1/2, - sqrt(3)/2) c. (sqrt(2)/2, -sqrt(2)/2) 56

57 Trig Functions and Rotations Find the six trig functions for = 5 /4 57

58 Solution Quadrant 3 cos = -1/sqrt(2) sin = -1/sqrt(2) tan = 1 sec = -sqrt(2) csc = -sqrt(2) cot = 1 58

59 Summary Can find points on the unit circle for angles in radians Can calculate trig functions for points on the unit circle 59

60 Chapter 3.4 Trigonometry of Real Numbers 60

61 Overview Define trig functions in terms of a real number t Find the number associated with special values of the trig functions Find the real number t associated with any trig value 61

62 Real Numbers Integers -2, -1, 0, 1, 2, Rationals: any number that can be expressed as a fraction Rationals include integers Reals: the non-imaginary numbers. Includes 2 and all rational numbers 62

63 Trigonometry of Real Numbers Work with the unit circle: r=1. if r = 1, then arc length s = r = That way, any function of is a function of arc length, s We can also treat reference arc as reference angle 63

64 Give the six trig functions of: a. 11 / 6 Examples b. 3 / 2 64

65 Solution a. Quadrant 4: cos = sqrt(3)/2 sin = -1/2 tan = -1/sqrt(3) b. Along y axis cos = 0 sin = -1 tan = undefined (- ) Given these, can find csc, sec, cot 65

66 csc ( /6) = More Examples csc (5 /6) = csc (11 /6) = csc (- /6) = csc (-17 /6) = 66

67 csc ( /6) = 2 Solutions csc (5 /6) = 2 ; /6 away from 180 csc (11 /6) = - 2 ; /6 away from 360 csc (- /6) = - 2 ; /6 away from 0 csc (-17 /6) = -2 ; /6 away from -18 /6 = -3 =

68 Examples cot = cot 0 = cot /2 = cot 3 /2 68

69 Solutions cot = undefined cot 0 =undefined cot /2 = 0 cot 3 /2 = 0 69

70 Special Angles Find t such that a. cos t = in quadrant 2 b. tan t = 3 in quadrant 3 70

71 Solutions Find t such that a. cos t = -1/sqrt(2); Q2 (like a 45 deg angle) t = 3 /4 b. tan t = sqrt (3); Q3(like a 60 deg angle) t = 4 /3 71

72 Summary Know how to find special angles on the unit circle Can calculate the trig functions of these angles 72

73 Chapter 3 Review 73

74 Give the reference angle for: 7 /6 24 /3 74

75 Give the reference angle for: Solution 7 /6 : / 6 24 /3: 8 or 0 75

76 Find Sin /3 = Cos 2 /3 = Tan /3 = Cos 7 /4 = Tan 7 /4 = 76

77 Solution 77

78 Verify that (1/3, - 2 sqrt(2) / 3) is a point on the unit circle. Find the value of all the trig functions associated with this point 78

79 Solution 79

80 A camera crew rids a cart on a circular arc. The radius of the arc is 75 ft and can sweep an angle of 172.5º in 20 sec. Find the length of the track in feet Find the angular velocity of the cart Find the linear velocity of the cart. 80

81 Solution A camera crew rids a cart on a circular arc. The radius of the arc is 75 ft and can sweep an angle of 172.5º in 20 sec. Find the length of the track in feet S=r = 75 ft (172.5 )( /180) = ft Find the angular velocity of the cart ω = /time = 172.5( /180) /20 = 0.15 rad/sec Find the linear velocity of the cart V = r ω = (0.15 rad/sec) (75ft) = ft/sec 81

82 Find t, between 0 and 2 if Sin t = -1/2 in Q3 Sec t = 2 sqrt(3)/3 in Q4 Tan t = -1 in Q2 82

83 Find t, between 0 and 2 if Sin t = -1/2 in Q3: t= 7 /6 Solution Sec t = 2 sqrt(3)/3 in Q4: t = 11 /6 Tan t = -1 in Q2: t = 3 /4 83

84 Example If (20/29, 21/29) is a point on the central unit circle, use symmetry to find 3 other points on the circle. 84

85 Solution If (20/29, 21/29) is a point on the central unit circle, use symmetry to find 3 other points on the circle. (-20/29, 21/29) (-20/29, -21/29) (20/29, -21/29) 85

86 Convert to radians:

87 Convert to radians: 300 : 5 / = -1.26; First change the minutes to seconds =

88 Convert to degrees: /

89 Solution Convert to degrees: 9.29 = /2 = = Remember, to convert to degrees, multiply by 180 / 89

90 Assume Memphis is in directly north of New Orleans, at 90 º W longitude. Find the distance between cities, in km, if the radius of the earth is 6000 km, Memphis is at 35º north, and New Orleans is at 29.6º. 90

91 Solution Assume Memphis is in directly north of New Orleans, at 90 º W longitude. Find the distance between cities, in km, if the radius of the earth is 6000 km, Memphis is at 35º north, and New Orleans is at 29.6º. Angle is 35º º = 5.4º 5.4º ( /180) = rad Radius is 6000 km, so arclength is 6000 ( 0.094) = 565 km 91

92 Chapter 3 Summary Test With additional Chapter 3 problems 92

93 1. What angles in [0, 2 ) make this true? Sin t = - 3 / 2 93

94 Solution 1. What angles in [0, 2 ) make this true? Sin t = - 3 / 2 t=7 /6, 11 /6 94

95 2. Given (3/4, - sqrt(7)/4) is a point on the unit circle, find all 6 trig functions. 95

96 Solution 2. Given (3/4, - sqrt(7)/4) is a point on the unit circle, find all 6 trig functions. x =3/4, y = -sqrt(7)/4, r = 1 cos = 3/4 sec = 4/3 sin = -sqrt(7)/4 tan = -sqrt(7)/3 csc = -4/sqrt(7) cot = -3/sqrt(7) 96

97 3. Find the complement of

98 Solution 3. Find the complement of ; Need 90 the angle. Also, 67 = need this to subtract

99 4. Given cot ( /8) = sqrt(2) + 1, find tan 2 (3 /8) 99

100 Solution 4. Given cot ( /8) = sqrt(2) + 1, find tan 2 (3 /8) Note that /8 + 3 /8 = /2. These are cofunctions! All that remains is to square [sqrt(2)+1] we get 2 sqrt(2)+3; since (a+b) 2 = a 2 + b 2 + 2ab 100

101 5. Given A = 9x, B = (6x+4), and C = 7x, find the measures of the angles A, B, C in triangle ABC 101

102 Solution 5. Given A = 9x, B = (6x+4), and C = 7x, find the measures of the angles A, B, C in triangle ABC 9x + 6x x = 180; the sum of the angles is x = x = 176/22 = 8 A = 72, B = 52, c = 56. Check =

103 6. You stand 457 m from a tower. The angle of elevation to the top of the tower is 30 deg. How tall is the tower? 103

104 Solution 6. You stand 457 m from a tower. The angle of elevation to the top of the tower is 30 deg. How tall is the tower? tan 30 = H/457, where H is height. tan 30 = 1/sqrt(3), so H = 457 sqrt(3) 104

105 7. One angle in a right triangle is , find the other two in radians 105

106 Solution 7. One angle in a right triangle is , find the other two One angle is a right angle, /2. The other two must sum to 90 Again, need to convert to decimal degrees, then calculate (90 angle) ( / 180) =

107 8. Find the value of the trig functions if the point (-5, -8) is on the terminal side of the angle 107

108 Solution 8. Find the value of the trig functions if the point (-5, -8) is on the terminal side of the angle. Pythagorean thm: = 89, hyp = sqrt (89) We are in quadrant 3, so only tan and cot are positive Cos = -5/sqrt(89) Sin = -8/sqrt(89) Tan = 8/5 108

109 10. t = / 6; find the trig functions 109

110 Solution 10. t = / 6; find the trig functions / 6 corresponds to a 30 deg angle. Cos = sqrt(3)/2 Sin = 1/2 Tan = 1/(sqrt(3)) From these you can get the other three 110

111 13. A conveyor belt moves on rollers 2 in radius, turning at 252 rpm. Find the angular velocity of the rollers. How fast are your groceries moving on the belt? 111

112 Solution 13. A conveyor belt moves on rollers 2 in radius, turning at 252 rpm. Find the angular velocity of the rollers. How fast are your groceries moving on the belt? Angular velocity = angle/time; 252 rpm is 252 (2 )/min linear velocity is just r (angular velocity) = 2(252)(2 )in/min 112

113 15. Verify (sin x cos x + cos x) / (sin x + sin 2 x) = cot x 113

114 Solution 15. Verify (sin x cos x + cos x) / (sin x + sin 2 x) = cot x Take a cos x out of the left numerator, and sin x out of the denominator: cos x (sin x + 1)/ [sin x (sin x + 1)] = cot x remove (sin x + 1) from top and bottom, leaving cos x / sin x = cot x 114

115 16. If t = 5.37, in what quadrant does it terminate? 115

116 Solution 16. If t = 5.37, in what quadrant does it terminate? 2 is t is close to that, and certainly greater than 3 /2, so is in the 4 th quadrant 116

117 18. Find the values of sin, cos, tan, if =

118 Solution 18. Find the values of sin, cos, tan, if = 225 Reference angle is = 45, Q 3. Cos = -1/sqrt(2) Sin = -1/sqrt(2) Tan = 1 118

119 23. Given that (sin A)(tanA) >0 and (cos 2 A)(sin A) < 0, in which quadrant is the terminal side of A? 119

120 Solution 23. Given that (sin A)(tanA) >0 and (cos 2 A)(sin A) < 0, in which quadrant is the terminal side of A? If ab > 0, the a and b are either both > 0 or both < 0. Therfore, Need sin and tan both > or both < 0, but (cos 2 A)> 0, which means that sin <0, so tan < 0. Sin and Tan are < 0 in Q4. 120