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# Lesson 16: Inverse Trigonometric Functions (handout)

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We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses …

We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.

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• 1. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes Sec on 3.6 Inverse Trigonometric Func ons V63.0121.001: Calculus I Professor Ma hew Leingang New York University March 28, 2011 . . Notes Announcements Midterm has been returned. Please see FAQ on Blackboard (under ”Exams and Quizzes”) Quiz 3 this week in recita on on Sec on 2.6, 2.8, 3.1, 3.2 Quiz 4 April 14–15 on Sec ons 3.3, 3.4, 3.5, and 3.7 Quiz 5 April 28–29 on Sec ons 4.1, 4.2, 4.3, and 4.4 . . Notes Objectives Know the deﬁni ons, domains, ranges, and other proper es of the inverse trignometric func ons: arcsin, arccos, arctan, arcsec, arccsc, arccot. Know the deriva ves of the inverse trignometric func ons. . . . 1.
• 2. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes Outline Inverse Trigonometric Func ons Deriva ves of Inverse Trigonometric Func ons Arcsine Arccosine Arctangent Arcsecant Applica ons . . Notes What is an inverse function? Deﬁni on Let f be a func on with domain D and range E. The inverse of f is the func on f−1 deﬁned by: f−1 (b) = a, where a is chosen so that f(a) = b. So f−1 (f(x)) = x, f(f−1 (x)) = x . . Notes What functions are invertible? In order for f−1 to be a func on, there must be only one a in D corresponding to each b in E. Such a func on is called one-to-one The graph of such a func on passes the horizontal line test: any horizontal line intersects the graph in exactly one point if at all. If f is con nuous, then f−1 is con nuous. . . . 2.
• 3. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes Graphing the inverse function y y=x If b = f(a), then f−1 (b) = a. So if (a, b) is on the graph of f, then (b, a) is on the graph of f−1 . On the xy-plane, the point (b, a) (b, a) is the reﬂec on of (a, b) in the line y = x. (a, b) Therefore: . x Fact The graph of f−1 is the reﬂec on of the graph of f in the line y = x. . . Notes arcsin Arcsin is the inverse of the sine func on a er restric on to [−π/2, π/2]. y y=x arcsin . x π π sin − 2 2 The domain of arcsin is [−1, 1] [ π π] The range of arcsin is − , 2 2 . . Notes arccos Arccos is the inverse of the cosine func on a er restric on to [0, π] arccos y y=x cos . x 0 π The domain of arccos is [−1, 1] The range of arccos is [0, π] . . . 3.
• 4. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes arctan y=x Arctan is the inverse of the tangent func on a er restric on to y (−π/2, π/2). π 2 arctan . x 3π π π 3π − − 2 2− π 2 2 2 ( π π ∞) The domain of arctan is (−∞, ) The range of arctan is − , 2 2 tan π π lim arctan x = , lim arctan x = − . x→∞ 2 x→−∞ 2 . Notes arcsec 3π 2 Arcsecant is the inverse of secant a er restric on to x y= [0, π/2) ∪ [π, 3π/2). y π 2 . x 3π π π 3π − − 2 2 2 2 The domain of arcsec is (−∞, −1] ∪ [1, ∞) [ π ) (π ] The range of arcsec is 0, ∪ ,π 2 2 π 3π lim arcsec x = , lim arcsec x = sec x→∞ 2 x→−∞ 2 . . Notes Values of Trigonometric Functions x 0 π/6 π/4 π/3 π/2 √ √ sin x 0 1/2 2/2 3/2 1 √ √ cos x 1 3/2 2/2 1/2 0 √ √ tan x 0 1/ 3 1 3 undef √ √ cot x undef 3 1 1/ 3 0 √ √ sec x 1 2/ 3 2/ 2 2 undef √ √ csc x undef 2 2/ 2 2/ 3 1 . . . 4.
• 5. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes Check: Values of inverse trigonometric functions Example Solu on Find π arcsin(1/2) 6 arctan(−1) ( √ ) 2 arccos − 2 . . Notes Caution: Notational ambiguity sin2 x =.(sin x)2 sin−1 x = (sin x)−1 sinn x means the nth power of sin x, except when n = −1! The book uses sin−1 x for the inverse of sin x, and never for (sin x)−1 . 1 I use csc x for and arcsin x for the inverse of sin x. sin x . . Notes Outline Inverse Trigonometric Func ons Deriva ves of Inverse Trigonometric Func ons Arcsine Arccosine Arctangent Arcsecant Applica ons . . . 5.
• 6. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes The Inverse Function Theorem Theorem (The Inverse Func on Theorem) Let f be diﬀeren able at a, and f′ (a) ̸= 0. Then f−1 is deﬁned in an open interval containing b = f(a), and 1 (f−1 )′ (b) = f′ (f−1 (b)) In Leibniz nota on we have dx 1 = dy dy/dx . . Notes Illustrating the IFT Example Use the inverse func on theorem to ﬁnd the deriva ve of the square root func on. Solu on (Newtonian nota on) √ Let f(x) = x2 so that f−1 (y) = y. Then f′ (u) = 2u so for any b > 0 we have 1 (f−1 )′ (b) = √ 2 b . . Notes Illustrating the IFT Example Use the inverse func on theorem to ﬁnd the deriva ve of the square root func on. Solu on (Leibniz nota on) If the original func on is y = x2 , then the inverse func on is deﬁned by x = y2 . Diﬀeren ate implicitly: dy dy 1 1 = 2y =⇒ = √ dx dx 2 x . . . 6.
• 7. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes The derivative of arcsine Let y = arcsin x, so x = sin y. Then dy dy 1 1 cos y = 1 =⇒ = = dx dx cos y cos(arcsin x) To simplify, look at a right triangle: √ cos(arcsin x) = 1 − x2 1 So x Fact d 1 y = arcsin x arcsin(x) = √ .√ dx 1 − x2 1 − x2 . . Notes Graphing arcsin and its derivative 1 √ The domain of f is [−1, 1], 1 − x2 but the domain of f′ is (−1, 1) arcsin lim− f′ (x) = +∞ x→1 | . | lim f′ (x) = +∞ −1 1 x→−1+ . . Notes Composing with arcsin Example Let f(x) = arcsin(x3 + 1). Find f′ (x). Solu on . . . 7.
• 8. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes The derivative of arccos Let y = arccos x, so x = cos y. Then dy dy 1 1 − sin y = 1 =⇒ = = dx dx − sin y − sin(arccos x) To simplify, look at a right triangle: √ sin(arccos x) = 1 − x2 1 √ So 1 − x2 Fact d 1 y = arccos x arccos(x) = − √ . dx 1 − x2 x . . Notes Graphing arcsin and arccos arccos Note (π ) cos θ = sin −θ arcsin 2 π =⇒ arccos x = − arcsin x 2 | . | −1 1 So it’s not a surprise that their deriva ves are opposites. . . Notes The derivative of arctan Let y = arctan x, so x = tan y. Then dy dy 1 sec2 y = 1 =⇒ = = cos2 (arctan x) dx dx sec2 y To simplify, look at a right triangle: 1 cos(arctan x) = √ 1 + x2 √ So 1 + x2 x Fact d 1 y = arctan x . arctan(x) = 1 dx 1 + x2 . . . 8.
• 9. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes Graphing arctan and its derivative y π/2 arctan 1 1 + x2 . x −π/2 The domain of f and f′ are both (−∞, ∞) Because of the horizontal asymptotes, lim f′ (x) = 0 x→±∞ . . Notes Composing with arctan Example √ Let f(x) = arctan x. Find f′ (x). Solu on . . Notes The derivative of arcsec Try this ﬁrst. . . . 9.
• 10. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Notes Another Example Example Let f(x) = earcsec 3x . Find f′ (x). Solu on . . Notes Outline Inverse Trigonometric Func ons Deriva ves of Inverse Trigonometric Func ons Arcsine Arccosine Arctangent Arcsecant Applica ons . . Notes Application Example One of the guiding principles of most sports is to “keep your eye on the ball.” In baseball, a ba er 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 ba er’s angle of gaze need to change to follow the ball as it crosses home plate? . . . 10.
• 11. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons . . March 28, 2011 Solu on Notes . . Notes Summary y y′ y y′ 1 1 arcsin x √ arccos x − √ 1−x 2 1 − x2 1 1 arctan x arccot x − 1 + x2 1 + x2 1 1 arcsec x √ arccsc x − √ x x2 − 1 x x2 − 1 Remarkable that the deriva ves of these transcendental func ons are algebraic (or even ra onal!) . . Notes . . . 11.