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. Newton's derivation of the sine and arcsine series Newton dis- covered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point. a. Referring to the figure, show that x = sin s or s = sin-!x. b. The area of a circular sector of radiusr subtended by an angle O is 1/2 r0. Show that the area of the circular sector APE is 1/2 1 s/2, which implies that := 2VT -² dt – xVī - x². c. Use the binomial series for f(x) = V1 - x² to obtain the first few terms of the Taylor series for s = sin!x. d. Newton next inverted the series in part (c) to obtain the Taylor series for x = sin s. He did this by assuming sin s = East and solving x = sin (sin-!x) for the coeffi- cients a. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well). fk) =VT -x? х

. Newton's derivation of the sine and arcsine series Newton dis- covered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point. a. Referring to the figure, show that x = sin s or s = sin-!x. b. The area of a circular sector of radiusr subtended by an angle O is 1/2 r0. Show that the area of the circular sector APE is 1/2 1 s/2, which implies that := 2VT -² dt – xVī - x². c. Use the binomial series for f(x) = V1 - x² to obtain the first few terms of the Taylor series for s = sin!x. d. Newton next inverted the series in part (c) to obtain the Taylor series for x = sin s. He did this by assuming sin s = East and solving x = sin (sin-!x) for the coeffi- cients a. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well). fk) =VT -x? х

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 74E
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. Newton's derivation of the sine and arcsine series Newton dis- covered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point. a. Referring to the figure, show that x = sin s or s = sin-!x. b. The area of a circular sector of radiusr subtended by an angle O is 1/2 r0. Show that the area of the circular sector APE is 1/2 1 s/2, which implies that := 2VT -² dt – xVī - x². c. Use the binomial series for f(x) = V1 - x² to obtain the first few terms of the Taylor series for s = sin!x. d. Newton next inverted the series in part (c) to obtain the Taylor series for x = sin s. He did this by assuming sin s = East and solving x = sin (sin-!x) for the coeffi- cients a. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well). fk) =VT -x? х
Transcribed Image Text:. Newton's derivation of the sine and arcsine series Newton dis- covered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point. a. Referring to the figure, show that x = sin s or s = sin-!x. b. The area of a circular sector of radiusr subtended by an angle O is 1/2 r0. Show that the area of the circular sector APE is 1/2 1 s/2, which implies that := 2VT -² dt – xVī - x². c. Use the binomial series for f(x) = V1 - x² to obtain the first few terms of the Taylor series for s = sin!x. d. Newton next inverted the series in part (c) to obtain the Taylor series for x = sin s. He did this by assuming sin s = East and solving x = sin (sin-!x) for the coeffi- cients a. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well). fk) =VT -x? х
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