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Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 9.4 - Problem 80ae
Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 9.4 - Problem 80ae

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# Newton's derivation of the sine and an sine series Newton

ISBN: 9780321570567 2

## Solution for problem 80AE Chapter 9.4

Calculus: Early Transcendentals | 1st Edition

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Problem 80AE

Newton's derivation of the sine and an sine series Newton discovered 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 y or y = sin?1x.b. The area of a circular" sector of radius r subtended by an angle ? is . Show that the area of the circular sector APE is y/2, which implies that .c. Use the binomial series for to obtain the first few terms of the Taylor series for y = sin?1 x.d. Newton next inverted the series in part (c) to obtain the Taylor series for x = sin y. He did ibis by assuming that sin = ?akyk and solving x = sin (sin?1 x)for the coefficients ak. Find the first few terms of the Taylor series for sin v using this idea (a computer algebra system might be helpful as well).

Step-by-Step Solution:
Step 1 of 3

Solution 80AEconsider the figure (a) from above figure, we have to show that x = sin y or y = sin-1 xConsider the triangle PBE, in the above figure From the triangle PBE, we have sin y = 1/xx = sin yOr y = sin-1 x(b) Area of a circular sector of radius r subtended by the angle is ½ r2 therefore , area of the circular sector APE = ½ 12 y since r =1 and = ½ yArea of a triangle PFE is ½ x since ½ x (base ) (height ) .

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##### ISBN: 9780321570567

Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. The full step-by-step solution to problem: 80AE from chapter: 9.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. This full solution covers the following key subjects: sin, Series, obtain, newton, Taylor. This expansive textbook survival guide covers 85 chapters, and 5218 solutions. Since the solution to 80AE from 9.4 chapter was answered, more than 336 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. The answer to “Newton's derivation of the sine and an sine series Newton discovered 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 y or y = sin?1x.b. The area of a circular" sector of radius r subtended by an angle ? is . Show that the area of the circular sector APE is y/2, which implies that .c. Use the binomial series for to obtain the first few terms of the Taylor series for y = sin?1 x.d. Newton next inverted the series in part (c) to obtain the Taylor series for x = sin y. He did ibis by assuming that sin = ?akyk and solving x = sin (sin?1 x)for the coefficients ak. Find the first few terms of the Taylor series for sin v using this idea (a computer algebra system might be helpful as well).” is broken down into a number of easy to follow steps, and 156 words.

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