## Calculating Sines

### January 12, 2010

[ Today’s exercise was written by guest author Bill Cruise, who blogs at Bill the Lizard, where he is currently describing his adventures studying SICP. Feel free to contact me if you have an idea for an exercise, or if you wish to contribute your own exercise. ]

We calculated the value of pi, and logarithms to seven digits, in two previous exercises. We continue that thread in the current exercise with a function that calculates sines.

Sines were discovered by the Indian astronomer Aryabhata in the sixth century, further developed by the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī (from whose name derives our modern word algorithm) in the ninth century. Sines were studied by European mathematicians Leibniz and Euler in the seventeenth and eighteenth centuries. It was Euler who coined the word “sine”, based on an earlier mis-translation (to the Latin “sinus”) of the word “jya” used by Aryabhata. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. (You may remember the mnemonic SOHCAHTOA if you’ve ever taken a course in trigonometry.)

One way to calculate the sine of an angle expressed in radians is by summing terms of the Taylor series: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots = \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!}$

Another method of computing the sine comes from the triple-angle formula $\sin x = 3 \sin\frac{x}{3} - 4 \sin^3\frac{x}{3}$. Since the limit $\lim_{x \rightarrow\ 0} \frac{\sin\ x}{x} = 1$, a recursion that drives x to zero can calculate the sine of x.

Your task is to write two functions to calculate the sine of an angle, one based on the Taylor series and the other based on the recursive formula. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.