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Lesson 16: Inverse Trigonometric Functions (Section 041 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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    Lesson 16: Inverse Trigonometric Functions (Section 041 handout) Lesson 16: Inverse Trigonometric Functions (Section 041 handout) Document Transcript

    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 Section 3.5 Notes Inverse Trigonometric Functions V63.0121.041, Calculus I New York University November 1, 2010 Announcements Midterm grades have been submitted Quiz 3 this week in recitation on Section 2.6, 2.8, 3.1, 3.2 Thank you for the evaluations Announcements Notes Midterm grades have been submitted Quiz 3 this week in recitation on Section 2.6, 2.8, 3.1, 3.2 Thank you for the evaluations V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 2 / 31 Objectives Notes Know the definitions, domains, ranges, and other properties of the inverse trignometric functions: arcsin, arccos, arctan, arcsec, arccsc, arccot. Know the derivatives of the inverse trignometric functions. V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 3 / 31 1
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 What is an inverse function? Notes Definition Let f be a function with domain D and range E . The inverse of f is the function 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 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 4 / 31 What functions are invertible? Notes In order for f −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 −1 is continuous. V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 5 / 31 Outline Notes Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 6 / 31 2
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 arcsin Notes Arcsin is the inverse of the sine function after restriction 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 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 7 / 31 arccos Notes Arccos is the inverse of the cosine function after restriction 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] V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 8 / 31 arctan y =x Notes Arctan is the inverse of the tangent yfunction after restriction to [−π/2, π/2]. π 2 arctan x 3π π π 3π − − 2 2 π 2 2 − 2 The domain of arctan is (−∞, ∞) π π The range of arctan is − , 2 2 tan π π lim arctan x = , lim arctan x = − x→∞ 2 x→−∞ 2 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 9 / 31 3
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 3π arcsec 2 y =x Notes y Arcsecant is the inverse of secant after restriction to [0, π/2) ∪ (π, 3π/2]. 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 sec π 3π lim arcsec x = , lim arcsec x = x→∞ 2 x→−∞ 2 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 10 / 31 Values of Trigonometric Functions Notes π π π π x 0 6 4 3 2 √ √ 1 2 3 sin x 0 1 2 2 2 √ √ 3 2 1 cos x 1 0 2 2 2 1 √ tan x 0 √ 1 3 undef 3 √ 1 cot x undef 3 1 √ 0 3 2 2 sec x 1 √ √ 2 undef 3 2 2 2 csc x undef 2 √ √ 1 2 3 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 11 / 31 Check: Values of inverse trigonometric functions Notes Example Find arcsin(1/2) arctan(−1) √ 2 arccos − 2 Solution π 6 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 12 / 31 4
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 Caution: Notational ambiguity Notes 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 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 15 / 31 Outline Notes Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 16 / 31 The Inverse Function Theorem Notes Theorem (The Inverse Function Theorem) Let f be differentiable 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)) Upshot: Many times the derivative of f −1 (x) can be found by implicit differentiation and the derivative of f : V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 17 / 31 5
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 Derivation: The derivative of arcsin Notes 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 x So d 1 arcsin(x) = √ y = arcsin x dx 1 − x2 1 − x2 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 18 / 31 Graphing arcsin and its derivative Notes 1 √ 1 − x2 The domain of f is [−1, 1], but the domain of f is arcsin (−1, 1) lim f (x) = +∞ x→1− | | lim f (x) = +∞ −1 1 x→−1+ V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 19 / 31 Composing with arcsin Notes Example Let f (x) = arcsin(x 3 + 1). Find f (x). Solution V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 20 / 31 6
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 Derivation: The derivative of arccos Notes 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 1 − x2 So d 1 y = arccos x arccos(x) = − √ dx 1 − x2 x V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 21 / 31 Graphing arcsin and arccos Notes arccos Note π arcsin cos θ = sin −θ 2 π =⇒ arccos x = − arcsin x 2 | | −1 1 So it’s not a surprise that their derivatives are opposites. V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 22 / 31 Derivation: The derivative of arctan Notes 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 1 + x2 x So d 1 arctan(x) = y = arctan x dx 1 + x2 1 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 23 / 31 7
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 Graphing arctan and its derivative Notes 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→±∞ V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 24 / 31 Composing with arctan Notes Example √ Let f (x) = arctan x. Find f (x). Solution V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 25 / 31 Derivation: The derivative of arcsec Notes Try this first. Let y = arcsec x, so x = sec y . Then dy dy 1 1 sec y tan y = 1 =⇒ = = dx dx sec y tan y x tan(arcsec(x)) To simplify, look at a right triangle: √ x2 − 1 tan(arcsec x) = 1 x x2 − 1 So d 1 arcsec(x) = √ y = arcsec x dx x x2 − 1 1 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 26 / 31 8
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 Another Example Notes Example Let f (x) = e arcsec 3x . Find f (x). Solution 1 f (x) = e arcsec 3x · ·3 3x (3x)2 − 1 3e arcsec 3x = √ 3x 9x 2 − 1 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 31 Outline Notes Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Arcsine Arccosine Arctangent Arcsecant Applications V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 28 / 31 Application Notes Example One of the guiding principles of most sports is to “keep your eye on the ball.” 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’s angle of gaze need to change to follow the ball as it crosses home plate? V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 29 / 31 9
    • V63.0121.041, Calculus I Section 3.5 : Inverse Trigonometric Functions November 1, 2010 Solution Notes V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 30 / 31 Summary Notes y y 1 arcsin x √ 1 − x2 1 arccos x −√ 1 − x2 Remarkable that the 1 derivatives of these arctan x transcendental functions are 1 + x2 1 algebraic (or even rational!) arccot x − 1 + x2 1 arcsec x √ x x2 − 1 1 arccsc x − √ x x2 − 1 V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 31 / 31 Notes 10