Newtons derivation of the sine and arcsine series Newton

Chapter 9, Problem 86

(choose chapter or problem)

Newtons derivation of the sine and arcsine 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 s or s = sin -1 x. b. The area of a circular sector of radius r subtended by an angle u is 12 r2 u. Show that the area of the circular sector APE is s>2, which implies that s = 2 L x 0 21 - t2 dt - x21 - x2. c. Use the binomial series for f 1x2 = 21 - x2 to obtain the first few terms of the Taylor series for s = sin-1 x. d. Newton next inverted the series in part (c) to obtain the Taylor series for x = sin s. He did this by assuming that sin s = aaksk and solving x = sin 1sin-1 x2 for the coefficients ak. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well).