Chapter 7, Problem 34P

34. The Taylor series expansion for sin x about x = 0 is given by:

$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^{57}}{7!}+\mathrm{....}={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{(-1)}^n}{(2n+1)!}{x}^{2n+1}}$

where x is in radians. Write a user-defined function that determines sin x using Taylor’s series expansion. For function name and arguments, use y=sinTay (x), where the input argument x is the angle in degrees and the output argument y is the value of sins, inside the user-defined function, use a loop for adding the terms of the Taylor series. If a_nis the nth term in the series, then the sum S_nof the n terms is S_n= S_n-1+ a_n. in each pass, calculate the estimated error F given by $E=\left|\frac{S_n-S_{n-1}}{S_{n-1}}\right|$ . Stop adding terms when $E\le 0.000001$ . Since $\sin \theta =\sin (\theta \pm 360n)$(n is an integer) write the user- defined function such that if the angle is larger than 360°, or smaller than -360°, then the Taylor series will be calculated using the smallest number of terms (using a value for x that is closest to 0).

Use sinTay for calculating:

(a) sin39° (h) sin205° (c) sin(-70°)

(c) sin 754° (e) sin 19,000° (d) sin(-748°)

Compare the values calculated using sinTay with the values obtained by using MATLAB’s built-in sind function.