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Lesson 15: Inverse Trigonometric Functions

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• 1. Section 3.5 Inverse Trigonometric Functions V63.0121.006/016, Calculus I March 11, 2010 Announcements Exams returned in recitation There is WebAssign due Tuesday March 23 and written HW due Thursday March 25 . . . . . .
• 2. Announcements Exams returned in recitation There is WebAssign due Tuesday March 23 and written HW due Thursday March 25 next quiz is Friday April 2 . . . . . .
• 3. What is an inverse function? De&#xFB01;nition Let f be a function with domain D and range E. The inverse of f is the function f&#x2212;1 de&#xFB01;ned by: f&#x2212;1 (b) = a, where a is chosen so that f(a) = b. . . . . . .
• 4. What is an inverse function? De&#xFB01;nition Let f be a function with domain D and range E. The inverse of f is the function f&#x2212;1 de&#xFB01;ned by: f&#x2212;1 (b) = a, where a is chosen so that f(a) = b. So f&#x2212;1 (f(x)) = x, f(f&#x2212;1 (x)) = x . . . . . .
• 5. What functions are invertible? In order for f&#x2212;1 to be a function, there must be only one a in D corresponding to each b in E. Such a function is called one-to-one The graph of such a function passes the horizontal line test: any horizontal line intersects the graph in exactly one point if at all. If f is continuous, then f&#x2212;1 is continuous. . . . . . .
• 6. Outline Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications . . . . . .
• 7. arcsin Arcsin is the inverse of the sine function after restriction to [&#x2212;&#x3C0;/2, &#x3C0;/2]. y . . . . x . &#x3C0; &#x3C0; s . in &#x2212; . . 2 2 . . . . . .
• 8. arcsin Arcsin is the inverse of the sine function after restriction to [&#x2212;&#x3C0;/2, &#x3C0;/2]. y . . . . . x . &#x3C0; &#x3C0; s . in &#x2212; . . . 2 2 . . . . . .
• 9. arcsin Arcsin is the inverse of the sine function after restriction to [&#x2212;&#x3C0;/2, &#x3C0;/2]. y . y . =x . . . . x . &#x3C0; &#x3C0; s . in &#x2212; . . . 2 2 . . . . . .
• 10. arcsin Arcsin is the inverse of the sine function after restriction to [&#x2212;&#x3C0;/2, &#x3C0;/2]. y . . . rcsin a . . . . x . &#x3C0; &#x3C0; s . in &#x2212; . . . 2 2 . The domain of arcsin is [&#x2212;1, 1] [ &#x3C0; &#x3C0;] The range of arcsin is &#x2212; , 2 2 . . . . . .
• 11. arccos Arccos is the inverse of the cosine function after restriction to [0, &#x3C0;] y . c . os . . x . 0 . . &#x3C0; . . . . . .
• 12. arccos Arccos is the inverse of the cosine function after restriction to [0, &#x3C0;] y . . c . os . . x . 0 . . &#x3C0; . . . . . . .
• 13. arccos Arccos is the inverse of the cosine function after restriction to [0, &#x3C0;] y . y . =x . c . os . . x . 0 . . &#x3C0; . . . . . . .
• 14. arccos Arccos is the inverse of the cosine function after restriction to [0, &#x3C0;] . . rccos a y . . c . os . . . x . 0 . . &#x3C0; . The domain of arccos is [&#x2212;1, 1] The range of arccos is [0, &#x3C0;] . . . . . .
• 15. arctan Arctan is the inverse of the tangent function after restriction to [&#x2212;&#x3C0;/2, &#x3C0;/2]. y . . x . 3&#x3C0; &#x3C0; &#x3C0; 3&#x3C0; &#x2212; . &#x2212; . . . 2 2 2 2 t .an . . . . . .
• 16. arctan Arctan is the inverse of the tangent function after restriction to [&#x2212;&#x3C0;/2, &#x3C0;/2]. y . . x . 3&#x3C0; &#x3C0; &#x3C0; 3&#x3C0; &#x2212; . &#x2212; . . . 2 2 2 2 t .an . . . . . .
• 17. arctan Arctan is the inverse of the tangent function after restriction to y . =x [&#x2212;&#x3C0;/2, &#x3C0;/2]. y . . x . 3&#x3C0; &#x3C0; &#x3C0; 3&#x3C0; &#x2212; . &#x2212; . . . 2 2 2 2 t .an . . . . . .
• 18. arctan Arctan is the inverse of the tangent function after restriction to [&#x2212;&#x3C0;/2, &#x3C0;/2]. y . &#x3C0; . a . rctan 2 . x . &#x3C0; &#x2212; . 2 The domain of arctan is (&#x2212;&#x221E;, &#x221E;) ( &#x3C0; &#x3C0;) The range of arctan is &#x2212; , 2 2 &#x3C0; &#x3C0; lim arctan x = , lim arctan x = &#x2212; x&#x2192;&#x221E; 2 x&#x2192;&#x2212;&#x221E; 2 . . . . . .
• 19. arcsec Arcsecant is the inverse of secant after restriction to [0, &#x3C0;/2) &#x222A; (&#x3C0;, 3&#x3C0;/2]. y . . x . 3&#x3C0; &#x3C0; &#x3C0; 3&#x3C0; &#x2212; . &#x2212; . . . 2 2 2 2 s . ec . . . . . .
• 20. arcsec Arcsecant is the inverse of secant after restriction to [0, &#x3C0;/2) &#x222A; (&#x3C0;, 3&#x3C0;/2]. y . . . x . 3&#x3C0; &#x3C0; &#x3C0; 3&#x3C0; &#x2212; . &#x2212; . . . . 2 2 2 2 s . ec . . . . . .
• 21. arcsec Arcsecant is the inverse of secant after restriction to y . =x [0, &#x3C0;/2) &#x222A; (&#x3C0;, 3&#x3C0;/2]. y . . . x . 3&#x3C0; &#x3C0; &#x3C0; 3&#x3C0; &#x2212; . &#x2212; . . . . 2 2 2 2 s . ec . . . . . .
• 22. arcsec 3&#x3C0; . Arcsecant is the inverse of secant after restriction to 2 [0, &#x3C0;/2) &#x222A; (&#x3C0;, 3&#x3C0;/2]. . . y &#x3C0; . 2 . . . x . . The domain of arcsec is (&#x2212;&#x221E;, &#x2212;1] &#x222A; [1, &#x221E;) [ &#x3C0; ) (&#x3C0; ] The range of arcsec is 0, &#x222A; ,&#x3C0; 2 2 &#x3C0; 3&#x3C0; lim arcsec x = , lim arcsec x = x&#x2192;&#x221E; 2 x&#x2192;&#x2212;&#x221E; 2 . . . . . .
• 23. Values of Trigonometric Functions &#x3C0; &#x3C0; &#x3C0; &#x3C0; x 0 6 4 3 2 &#x221A; &#x221A; 1 2 3 sin x 0 1 2 2 2 &#x221A; &#x221A; 3 2 1 cos x 1 0 2 2 2 1 &#x221A; tan x 0 &#x221A; 1 3 undef 3 &#x221A; 1 cot x undef 3 1 &#x221A; 0 3 2 2 sec x 1 &#x221A; &#x221A; 2 undef 3 2 2 2 csc x undef 2 &#x221A; &#x221A; 1 2 3 . . . . . .
• 24. Check: Values of inverse trigonometric functions Example Find arcsin(1/2) arctan(&#x2212;1) ( &#x221A; ) 2 arccos &#x2212; 2 . . . . . .
• 25. Check: Values of inverse trigonometric functions Example Find arcsin(1/2) arctan(&#x2212;1) ( &#x221A; ) 2 arccos &#x2212; 2 Solution &#x3C0; 6 . . . . . .
• 26. What is arctan(&#x2212;1)? . 3 . &#x3C0;/4 . . . . &#x2212; . &#x3C0;/4 . . . . . .
• 27. What is arctan(&#x2212;1)? . ( ) 3 . &#x3C0;/4 3&#x3C0; . Yes, tan = &#x2212;1 4 &#x221A; 2 s . in(3&#x3C0;/4) = 2 . &#x221A; . 2 . os(3&#x3C0;/4) = &#x2212; c 2 . &#x2212; . &#x3C0;/4 . . . . . .
• 28. What is arctan(&#x2212;1)? . ( ) 3 . &#x3C0;/4 3&#x3C0; . Yes, tan = &#x2212;1 4 &#x221A; But, the range of arctan ( &#x3C0; &#x3C0;) 2 s . in(3&#x3C0;/4) = is &#x2212; , 2 2 2 . &#x221A; . 2 . os(3&#x3C0;/4) = &#x2212; c 2 . &#x2212; . &#x3C0;/4 . . . . . .
• 29. What is arctan(&#x2212;1)? . ( ) 3 . &#x3C0;/4 3&#x3C0; . Yes, tan = &#x2212;1 4 But, the range of arctan ( &#x3C0; &#x3C0;) &#x221A; is &#x2212; , 2 2 2 c . os(&#x3C0;/4) = . 2 Another angle whose . &#x3C0; tangent is &#x2212;1 is &#x2212; , and &#x221A; 4 2 this is in the right range. . in(&#x3C0;/4) = &#x2212; s 2 . &#x2212; . &#x3C0;/4 . . . . . .
• 30. What is arctan(&#x2212;1)? . ( ) 3 . &#x3C0;/4 3&#x3C0; . Yes, tan = &#x2212;1 4 But, the range of arctan ( &#x3C0; &#x3C0;) &#x221A; is &#x2212; , 2 2 2 c . os(&#x3C0;/4) = . 2 Another angle whose . &#x3C0; tangent is &#x2212;1 is &#x2212; , and &#x221A; 4 2 this is in the right range. . in(&#x3C0;/4) = &#x2212; s &#x3C0; 2 So arctan(&#x2212;1) = &#x2212; 4 . &#x2212; . &#x3C0;/4 . . . . . .
• 31. Check: Values of inverse trigonometric functions Example Find arcsin(1/2) arctan(&#x2212;1) ( &#x221A; ) 2 arccos &#x2212; 2 Solution &#x3C0; 6 &#x3C0; &#x2212; 4 . . . . . .
• 32. Check: Values of inverse trigonometric functions Example Find arcsin(1/2) arctan(&#x2212;1) ( &#x221A; ) 2 arccos &#x2212; 2 Solution &#x3C0; 6 &#x3C0; &#x2212; 4 3&#x3C0; 4 . . . . . .
• 33. Caution: Notational ambiguity . in2 x =.(sin x)2 s . in&#x2212;1 x = (sin x)&#x2212;1 s sinn x means the nth power of sin x, except when n = &#x2212;1! The book uses sin&#x2212;1 x for the inverse of sin x, and never for (sin x)&#x2212;1 . 1 I use csc x for and arcsin x for the inverse of sin x. sin x . . . . . .
• 34. Outline Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications . . . . . .
• 35. Theorem (The Inverse Function Theorem) Let f be differentiable at a, and f&#x2032; (a) &#x338;= 0. Then f&#x2212;1 is de&#xFB01;ned in an open interval containing b = f(a), and 1 (f&#x2212;1 )&#x2032; (b) = &#x2032; &#x2212;1 f (f (b)) . . . . . .
• 36. Theorem (The Inverse Function Theorem) Let f be differentiable at a, and f&#x2032; (a) &#x338;= 0. Then f&#x2212;1 is de&#xFB01;ned in an open interval containing b = f(a), and 1 (f&#x2212;1 )&#x2032; (b) = &#x2032; &#x2212;1 f (f (b)) &#x201C;Proof&#x201D;. If y = f&#x2212;1 (x), then f(y ) = x , So by implicit differentiation dy dy 1 1 f&#x2032; (y) = 1 =&#x21D2; = &#x2032; = &#x2032; &#x2212;1 dx dx f (y) f (f (x)) . . . . . .
• 37. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =&#x21D2; = = dx dx cos y cos(arcsin x) . . . . . .
• 38. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =&#x21D2; = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: . . . . . . .
• 39. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =&#x21D2; = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: 1 . x . . . . . . . .
• 40. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =&#x21D2; = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: 1 . x . y . = arcsin x . . . . . . .
• 41. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =&#x21D2; = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: 1 . x . y . = arcsin x . &#x221A; . 1 &#x2212; x2 . . . . . .
• 42. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =&#x21D2; = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: &#x221A; cos(arcsin x) = 1 &#x2212; x2 1 . x . y . = arcsin x . &#x221A; . 1 &#x2212; x2 . . . . . .
• 43. The derivative of arcsin Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =&#x21D2; = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: &#x221A; cos(arcsin x) = 1 &#x2212; x2 1 . x . So d 1 y . = arcsin x arcsin(x) = &#x221A; dx 1 &#x2212; x2 . &#x221A; . 1 &#x2212; x2 . . . . . .
• 44. Graphing arcsin and its derivative 1 .&#x221A; 1 &#x2212; x2 The domain of f is [&#x2212;1, 1], but the domain . . rcsin a of f&#x2032; is (&#x2212;1, 1) lim f&#x2032; (x) = +&#x221E; x &#x2192;1 &#x2212; lim f&#x2032; (x) = +&#x221E; . | . . | x&#x2192;&#x2212;1+ &#x2212; . 1 1 . . . . . . . .
• 45. The derivative of arccos Let y = arccos x, so x = cos y. Then dy dy 1 1 &#x2212; sin y = 1 =&#x21D2; = = dx dx &#x2212; sin y &#x2212; sin(arccos x) . . . . . .
• 46. The derivative of arccos Let y = arccos x, so x = cos y. Then dy dy 1 1 &#x2212; sin y = 1 =&#x21D2; = = dx dx &#x2212; sin y &#x2212; sin(arccos x) To simplify, look at a right triangle: &#x221A; sin(arccos x) = 1 &#x2212; x2 1 . &#x221A; . 1 &#x2212; x2 So d 1 y . = arccos x arccos(x) = &#x2212; &#x221A; . dx 1 &#x2212; x2 x . . . . . . .
• 47. Graphing arcsin and arccos . . rccos a . . rcsin a . | . |. . &#x2212; . 1 1 . . . . . . . .
• 48. Graphing arcsin and arccos . . rccos a Note (&#x3C0; ) cos &#x3B8; = sin &#x2212;&#x3B8; . . rcsin a 2 &#x3C0; =&#x21D2; arccos x = &#x2212; arcsin x 2 . | . |. . So it&#x2019;s not a surprise that their &#x2212; . 1 1 . derivatives are opposites. . . . . . . .
• 49. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =&#x21D2; = = cos2 (arctan x) dx dx sec2 y . . . . . .
• 50. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =&#x21D2; = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: . . . . . . .
• 51. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =&#x21D2; = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: x . . 1 . . . . . . .
• 52. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =&#x21D2; = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: x . y . = arctan x . 1 . . . . . . .
• 53. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =&#x21D2; = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: &#x221A; . 1 + x2 x . y . = arctan x . 1 . . . . . . .
• 54. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =&#x21D2; = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: 1 cos(arctan x) = &#x221A; 1 + x2 &#x221A; . 1 + x2 x . y . = arctan x . 1 . . . . . . .
• 55. The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =&#x21D2; = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: 1 cos(arctan x) = &#x221A; 1 + x2 &#x221A; . 1 + x2 x . So d 1 y . = arctan x arctan(x) = . dx 1 + x2 1 . . . . . . .
• 56. Graphing arctan and its derivative y . . /2 &#x3C0; a . rctan 1 . 1 + x2 . x . &#x2212; . &#x3C0;/2 The domain of f and f&#x2032; are both (&#x2212;&#x221E;, &#x221E;) Because of the horizontal asymptotes, lim f&#x2032; (x) = 0 x&#x2192;&#xB1;&#x221E; . . . . . .
• 57. Example &#x221A; Let f(x) = arctan x. Find f&#x2032; (x). . . . . . .
• 58. Example &#x221A; Let f(x) = arctan x. Find f&#x2032; (x). Solution d &#x221A; 1 d&#x221A; 1 1 arctan x = (&#x221A; )2 x= &#xB7; &#x221A; dx 1+ x dx 1+x 2 x 1 = &#x221A; &#x221A; 2 x + 2x x . . . . . .
• 59. The derivative of arcsec Try this &#xFB01;rst. . . . . . .
• 60. The derivative of arcsec Try this &#xFB01;rst. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =&#x21D2; = = dx dx sec y tan y x tan(arcsec(x)) . . . . . .
• 61. The derivative of arcsec Try this &#xFB01;rst. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =&#x21D2; = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: . . . . . . .
• 62. The derivative of arcsec Try this &#xFB01;rst. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =&#x21D2; = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: . . . . . . .
• 63. The derivative of arcsec Try this &#xFB01;rst. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =&#x21D2; = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: x . . 1 . . . . . . .
• 64. The derivative of arcsec Try this &#xFB01;rst. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =&#x21D2; = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: x . y . = arcsec x . 1 . . . . . . .
• 65. The derivative of arcsec Try this &#xFB01;rst. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =&#x21D2; = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: &#x221A; x2 &#x2212; 1 tan(arcsec x) = &#x221A; 1 x . . x2 &#x2212; 1 y . = arcsec x . 1 . . . . . . .
• 66. The derivative of arcsec Try this &#xFB01;rst. Let y = arcsec x, so x = sec y. Then dy dy 1 1 sec y tan y = 1 =&#x21D2; = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: &#x221A; x2 &#x2212; 1 tan(arcsec x) = &#x221A; 1 x . . x2 &#x2212; 1 So d 1 y . = arcsec x arcsec(x) = &#x221A; . dx x x2 &#x2212; 1 1 . . . . . . .
• 67. Another Example Example Let f(x) = earcsec x . Find f&#x2032; (x). . . . . . .
• 68. Another Example Example Let f(x) = earcsec x . Find f&#x2032; (x). Solution 1 f&#x2032; (x) = earcsec x &#xB7; &#x221A; x x2 &#x2212; 1 . . . . . .
• 69. Outline Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications . . . . . .
• 70. Application Example One of the guiding principles of most sports is to &#x201C;keep your eye on the ball.&#x201D; In baseball, a batter stands 2 ft away from home plate as a pitch is thrown with a velocity of 130 ft/sec (about 90 mph). At what rate does the batter&#x2019;s angle of gaze need to change to follow the ball as it crosses home plate? . . . . . .
• 71. Let y(t) be the distance from the ball to home plate, and &#x3B8; the angle the batter&#x2019;s eyes make with home plate while following the ball. We know y&#x2032; = &#x2212;130 and we want &#x3B8;&#x2032; at the moment that y = 0. y . 1 . 30 ft/sec . &#x3B8; . . 2 . ft . . . . . .
• 72. Let y(t) be the distance from the ball to home plate, and &#x3B8; the angle the batter&#x2019;s eyes make with home plate while following the ball. We know y&#x2032; = &#x2212;130 and we want &#x3B8;&#x2032; at the moment that y = 0. We have &#x3B8; = arctan(y/2). Thus d&#x3B8; 1 1 dy = &#xB7; 2 2 dt dt 1 + ( y /2 ) y . 1 . 30 ft/sec . &#x3B8; . . 2 . ft . . . . . .
• 73. Let y(t) be the distance from the ball to home plate, and &#x3B8; the angle the batter&#x2019;s eyes make with home plate while following the ball. We know y&#x2032; = &#x2212;130 and we want &#x3B8;&#x2032; at the moment that y = 0. We have &#x3B8; = arctan(y/2). Thus d&#x3B8; 1 1 dy = &#xB7; 2 2 dt dt 1 + ( y /2 ) When y = 0 and y&#x2032; = &#x2212;130, y . then d&#x3B8; 1 1 = &#xB7; (&#x2212;130) = &#x2212;65 rad/sec 1 . 30 ft/sec dt y =0 1+0 2 . &#x3B8; . . 2 . ft . . . . . .
• 74. Let y(t) be the distance from the ball to home plate, and &#x3B8; the angle the batter&#x2019;s eyes make with home plate while following the ball. We know y&#x2032; = &#x2212;130 and we want &#x3B8;&#x2032; at the moment that y = 0. We have &#x3B8; = arctan(y/2). Thus d&#x3B8; 1 1 dy = &#xB7; 2 2 dt dt 1 + ( y /2 ) When y = 0 and y&#x2032; = &#x2212;130, y . then d&#x3B8; 1 1 = &#xB7; (&#x2212;130) = &#x2212;65 rad/sec 1 . 30 ft/sec dt y =0 1+0 2 . &#x3B8; The human eye can only . track at 3 rad/sec! . 2 . ft . . . . . .
• 75. Recap y y&#x2032; 1 arcsin x &#x221A; 1 &#x2212; x2 1 arccos x &#x2212; &#x221A; Remarkable that the 1 &#x2212; x2 derivatives of these 1 transcendental functions arctan x 1 + x2 are algebraic (or even 1 rational!) arccot x &#x2212; 1 + x2 1 arcsec x &#x221A; x x2 &#x2212; 1 1 arccsc x &#x2212; &#x221A; x x2 &#x2212; 1 . . . . . .