Functions Modeling Change A Preparation for Calculus Third Edition
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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 TRIGONOMETRIC FUNCTIONS SECTION 6.7 INVERSE TRIGONOMETRIC FUNCTIONS
3 Suppose we want to find the solution to: cos t.4 Page 285 Example #1 3
4 Let's graph the following on our calculator: y1 = cos(x) and y2 =.4 Page 285 Example #1 Window Value Xmin -3π/2 Xmax 5π/2 Xscl π/2 Ymin -1 Ymax 1 Yscl.1 4
5 Let's graph the following on our calculator: y1 = cos(x) and y2 = Page 285 Example # x
6 2nd Trace 5 (intersect) Procedure: x = , , , Page 285 Example # x
7 2nd Trace 5 (intersect) Procedure: x = , , , Page 285 Example #1 7
8 cos t.4 Means: "Find an angle whose cosine =.4" Page 285 Example #1 8
9 Alternate Procedure: cos t.4 We will use: "Inverse cosine" cos 1 t Page 285 Example #1 9
10 cos t.4 Output Input t Output cos 1 (.4) Input Page 285 Example #1 10
11 t cos 1 (.4) In radian mode: TI: 2nd cos.4) ENTER gives: 1 cos (.4) Page 286 Example #1 11
12 In other words: 1 cos (.4) Means: cos( ).4 Page 286 Example #1 12
13 In other words: 1 cos (.4) Means: cos( ).4 Page 286 Example #1 BUT... 13
14 This is only 1 solution: 1 cos (.4) Page 286 Example #1 14
15 This is only 1 solution: 1 cos (.4) What about the other solutions? Page 286 Example #1 15
16 This is only 1 solution: 1 cos (.4) Page 286 Example #1 16
17 What about: Page 286 Example #1 17
18 There are other solutions, which we can find using the symmetry of the cosine graph. Since t 1 = 1.159, we see that t 0 = because the cosine function is symmetric about the y-axis. Page 286 Example #1 18
19 In addition, the arch of the cosine graph from π/2 to π/2 is exactly the same shape as the arch from 3π/2 to 5π/2, so Page 286 Example #1 19
20 t2 = 2π = t3 = 2π = Page 286 Example #1 20
21 Use the inverse cosine to estimate when the rabbit population, R, reaches 12,000. R 5000cos t What do we do? Page 286 Example #2 21
22 Use the inverse cosine to estimate when the rabbit population, R, reaches 12, cos t cos t cos t Page 286 Example #2 cos t
23 We want the angle whose cosine is cos t t 6 1 cos (.4) 1 cos (.4) t Page 286 Example #2 23
24 We now finish our calculations: 6 t t t t 6 ( ) Page 286 Example #2 Are we done? 24
25 Summary of Transformations For the sinusoidal functions y acos( B( t h)) k a is the amplitude 2π/ B is the period h is the horizontal shift y = k is the midline B /2π is the frequency; that is, the number of cycles completed in unit time. Page 272 Blue Box 25
26 y acos( B( t h)) k y 5000cos t Page 286 Example #2 a is the amplitude B is the # of cycles in 2π 2π/ B is the period h is the horizontal shift y = k is the midline B /2π is the frequency = the number of cycles completed in unit time 26
27 y acos( B( t h)) k y 5000cos t Page 286 Example #2 a is the amplitude = 5000 B is the # of cycles in 2π π/6 2π/ B is the period = (2π)/(π/6) = 12 h is the horizontal shift = 0 y = k is the midline = B /2π is the frequency = the number of cycles completed in unit time = 1/12 27
28 y 5000cos t = vertical stretch (y=kf(x): k>1) - = reflection across the x axis π/6 = horizontal stretch (y=f(kx): k<1) = vertical shift (k>0: up) Page 286 Example #2 28
29 Let's use our calculator and graph: y1 5000cos t Page 286 Example #2 Window Value Xmin 0 Xmax 24 Xscl 12 Ymin 0 Ymax Yscl
30 Now add: y Page 286 Example #2 Window Value Xmin 0 Xmax 24 Xscl 12 Ymin 0 Ymax Yscl
31 Page 286 Example #2 31
32 Recall: a function can only have 1 output for each input. Page
33 Recall: a function can only have 1 output for each input. Example: sin(0 O ) = 0 sin(180 O ) = 0 This is okay... Page
34 Recall: a function can only have 1 output for each input. Example: sin(0 O ) = 0 sin(0 O ) = 0.5 This is NOT okay... Page
35 This equation: cos t.4 Page 285 Example #1 35
36 yielded many solutions: x = , , , , Page 285 Example # x
37 However, the inverse cosine key on a calculator gives only one of them, namely: 1 cos (.4) Why does the calculator select this particular solution? Page
38 From the graph below, we see that the angles in the interval 0 t π produce all values in the range of cos t once and once only. Page
39 The inverse cosine function inputs values of y between 1 and 1 (all possible values of cos t) and outputs angles between 0 and π. Page
40 We interpret the value of cos 1 (y) as the angle between 0 and π whose cosine is y. Page
41 Because an angle in radians determines an arc of the same measure on a unit circle, the inverse cosine of y is sometimes called the arccosine of y. Page
42 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 42
43 We summarize: The inverse cosine function, also called the arccosine function, is denoted by cos 1 y or arccos y. t = cos 1 y provided that y = cos t and 0 t π Page 287 Blue Box 43
44 We summarize: The inverse cosine function, also called the arccosine function, is denoted by cos 1 y or arccos y. t = cos 1 y provided that y = cos t and 0 t π In other words, if t = arccos y, then t is the angle between 0 and π whose cosine is y. Page 287 Blue Box 44
45 We summarize: The inverse cosine function, also called the arccosine function, is denoted by cos 1 y or arccos y. t = cos 1 y provided that y = cos t and 0 t π In other words, if t = arccos y, then t is the angle between 0 and π whose cosine is y. The domain of the inverse cosine is 1 y 1 and its range is 0 t π. Page 287 Blue Box 45
46 Evaluate (a) cos 1 (0) (b) arccos(1) (c) cos 1 ( 1) Page 288 Example #3 46
47 (a) cos 1 (0) means the angle between 0 and π whose cosine is 0. Since cos(π/2) = 0, we have cos 1 (0) = π/2. (b) means the angle between 0 and π whose cosine is 1. Since cos(0) = 1, we have arccos(1)=0. (c) cos 1 ( 1) means the angle between 0 and π whose cosine is 1. Since cos(π) = 1, we have cos 1 ( 1) = π. Page 288 Example #3 47
48 Warning! It is important to realize that the notation cos 1 y does not indicate the reciprocal of cos y. In other words, cos 1 y is not the same as (cos y) 1. For example, But 1 cos (0), since cos cos0 1 cos0 1 Page
49 The Inverse Sine and Inverse Tangent Functions The inverse sine function, also called the arcsine function, is denoted by sin 1 y or arcsin y. We define t = sin 1 y provided y = sin t and π/2 t π/2. The domain of the inverse sine is 1 y 1 and the range is π/2 t π/2. Page 289 Blue Box 49
50 The Inverse Sine and Inverse Tangent Functions The inverse tangent function, also called the arctangent function, is denoted by tan 1 y or arctan y. We define t = tan 1 y provided y = tan t and π/2 t π/2. The domain of the inverse tangent is < y < and the range is π/2 < t < π/2. Page 289 Blue Box 50
51 Solving Equations Using Reference Angles Because of the symmetry of the unit circle, the values of sine, cosine, and tangent of angles in the first quadrant can be used to find values of these functions for angles in the other three quadrants. Page
52 Use the values of the sine and cosine of 65 to find the sine and cosine of 65, 245, and 785. Page 290 Example #7 52
53 Use the values of the sine and cosine of 65 to find the sine and cosine of 65. Remember: cos θ = x and sin θ=y Let P = (cos 65, sin 65 ) = (0.422, 0.906) be the point on the unit circle given by the angle 65. Page 290 Example #7 53
54 we see that 65 gives a point labeled Q that is the reflection of P across the x-axis. Page 290 Ex #7 54
55 Thus, the y-coordinate of Q is the negative of the y-coordinate of P, so Q = (0.422, 0.906). Page 290 Ex #7 55
56 O O sin( 65 ) and cos( 65 )=0.422 Page 290 Ex #7 56
57 Use the values of the sine and cosine of 65 to find the sine and cosine of 245. Remember: cos θ = x and sin θ=y We see that 245 = gives point R that is diametrically opposite the point P. Page 290 Example #7 57
58 The coordinates of R are the negatives of the coordinates of P, so R = ( 0.422, 0.906). Page 290 Ex #7 58
59 O O sin(245 ) and cos(245 )= Page 290 Ex #7 59
60 Use the values of the sine and cosine of 65 to find the sine and cosine of 785. Remember: cos θ = x and sin θ=y Since 785 = P = (cos 785, sin 785 ) = (0.422, 0.906) be the point on the unit circle given by the angle 785, which is the same point (P) as 65. Page 290 Example #7 60
61 O O sin(785 ) and cos(785 )=0.422 Page 290 Ex #7 61
62 In this example, we call 65 the reference angle. For an angle θ corresponding to the point P on the unit circle, the reference angle of θ is the angle between the line joining P to the origin and the nearest part of the x- axis. A reference angle is always between 0 and 90 ; that is, between 0 and π/2. Page
63 Page
64 Page
65 Use reference angles to solve the equations: (a) cos θ = for 0 θ 360. (b) tan θ = for 0 θ 720. Page 291 Example #8 65
66 Use reference angles to solve the equations: (a) cos θ = for 0 θ 360. Step 1: cos 1 (0.422) =? (Use degree mode on your calculator.) Page 291 Example #8 66
67 Use reference angles to solve the equations: (a) cos θ = for 0 θ 360. Step 1: cos 1 (0.422) = 65 Page 291 Example #8 67
68 Use reference angles to solve the equations: (a) cos θ = for 0 θ 360. Step 1: cos 1 (0.422) = 65 Step 2: in which quadrants is cos +? Page 291 Example #8 68
69 Use reference angles to solve the equations: (a) cos θ = for 0 θ 360. Step 1: cos 1 (0.422) = 65 Step 2: in which quadrants is cos +? I, IV Page 291 Example #8 69
70 Use reference angles to solve the equations: (a) cos θ = for 0 θ 360. Step 1: cos 1 (0.422) = 65 Step 2: in which quadrants is cos +? I, IV Step 3: what angle in quadrant IV has 65 as a reference angle? Page 291 Example #8 70
71 Use reference angles to solve the equations: (a) cos θ = for 0 θ 360. Step 1: cos 1 (0.422) = 65 Step 2: in which quadrants is cos +? I, IV Step 3: what angle in quadrant IV has 65 as a reference angle? IV: = 295 Page 291 Example #8 71
72 Use reference angles to solve the equations: (a) cos θ = for 0 θ 360. Step 1: cos 1 (0.422) = 65 Step 2: in which quadrants is cos +? I, IV Step 3: what angle in quadrant IV has 65 as a reference angle? IV: = 295 Therefore: 65, 295. Page 291 Example #8 72
73 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Page 291 Example #8 73
74 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Step 1: tan 1 (2.145) =? (Use degree mode on your calculator.) Page 291 Example #8 74
75 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Step 1: tan 1 (2.145) = 65 Page 291 Example #8 75
76 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Step 1: tan 1 (2.145) = 65 Step 2: in which quadrants is tan +? Page 291 Example #8 76
77 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Step 1: tan 1 (2.145) = 65 Step 2: in which quadrants is tan +? I, III Page 291 Example #8 77
78 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Step 1: tan 1 (2.145) = 65 Step 2: in which quadrants is tan +? I, III Step 3: what angle in quadrant III has 65 as a reference angle? Page 291 Example #8 78
79 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Step 1: tan 1 (2.145) = 65 Step 2: in which quadrants is tan +? I, III Step 3: what angle in quadrant III has 65 as a reference angle? IV: = 245 Page 291 Example #8 79
80 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Step 1: tan 1 (2.145) = 65 Step 2: in which quadrants is tan +? I, III Step 3: what angle in quadrant III has 65 as a reference angle? IV: = 245 Therefore: 65,245. Page 291 Example #8 80
81 Use reference angles to solve the equations: (b) tan θ = for 0 θ 720. Step 1: tan 1 (2.145) = 65 Step 2: in which quadrants is tan +? I, III Step 3: what angle in quadrant III has 65 as a reference angle? IV: Therefore: 65, 245, 425, 605. Page 291 Example #8 81
82 End of Section
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