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).

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 ) .