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(a) Prove the Calculus identity sin- (a) + sin-(VT- 2²) = 5, z€ (0,1], where sin-, denotes the usual real inverse trig functions.

(a) Prove the Calculus identity sin- (a) + sin-(VT- 2²) = 5, z€ (0,1], where sin-, denotes the usual real inverse trig functions.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.4: Multiple-angle Formulas
Problem 41E
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(a) Prove the Caleulus identity sin-(x) + sin-(V1– 2²) = 5 r€ (0, 1), %3D where sin-, denotes the usual real inverse trig functions. (b) Create and submit a plot of sin (r)+ sin-(VT- r²). Explain why the identity is false if a < 0. (c) In this question, all integrals are Calculus-type real integrals. Let f(t) = VT-t,-1sts1, and consider the substitution s = f(t). Taking a derivative gives tdt ds = - Hence, dt ds By the substitution rule for definite integrals, dt ds dt dt Vī dt dt dt %3D VI- 12 0. However, as in (b), the identity is false if r < 0. Explain what went wrong. (d) We now move off the real line and into the complex plane. Suppose wi + w = 1 and that wi, w2 e RHP (open right half-plane). Let C1, C2 be curves that, respectively, connect 0 to w, and w, and that avoid K = (-00,-1]U[1, 00). Use the substitution rule for complex line integrals to show that, with appropriately chose branches of (1 – 22)/2, we have %3D dç Jo, (1- (?)1/2 + Jc, 1 – (*)!/2 = 7 (1- C2)1/2 = 2
Transcribed Image Text:(a) Prove the Caleulus identity sin-(x) + sin-(V1– 2²) = 5 r€ (0, 1), %3D where sin-, denotes the usual real inverse trig functions. (b) Create and submit a plot of sin (r)+ sin-(VT- r²). Explain why the identity is false if a < 0. (c) In this question, all integrals are Calculus-type real integrals. Let f(t) = VT-t,-1sts1, and consider the substitution s = f(t). Taking a derivative gives tdt ds = - Hence, dt ds By the substitution rule for definite integrals, dt ds dt dt Vī dt dt dt %3D VI- 12 0. However, as in (b), the identity is false if r < 0. Explain what went wrong. (d) We now move off the real line and into the complex plane. Suppose wi + w = 1 and that wi, w2 e RHP (open right half-plane). Let C1, C2 be curves that, respectively, connect 0 to w, and w, and that avoid K = (-00,-1]U[1, 00). Use the substitution rule for complex line integrals to show that, with appropriately chose branches of (1 – 22)/2, we have %3D dç Jo, (1- (?)1/2 + Jc, 1 – (*)!/2 = 7 (1- C2)1/2 = 2
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