TITLE: How to calculate the sine manually, without any rules, calculator or anything else?
QUESTION [7 upvotes]: I want to know how to calculate the value of sin, not using table values or calculator.
I found this $\frac{(e^{ix})^2-1}{2ie^{ix}}$, but how to deal with $i$ number, if it's $\sqrt{-1}$?

REPLY [5 votes]: For a remarkably good approximation with relatively little effort, consider Bhaskara I's sine approximation formula. For angles in degrees (between 0 and 180), this takes the form
$$
\sin x^\circ \approx \frac{4x(180-x)}{40500-x(180-x)}
$$
while in radians, it's
$$
\sin x \approx \frac{16x(\pi-x)}{5\pi^2-4x(\pi-x)}=\frac{16\frac{x}\pi(1-\frac{x}\pi)}{5-4\frac{x}\pi(1-\frac{x}\pi)}
$$
This expression has a relative error of less than 2%, which occurs for very small (and for very close to 180$^\circ$) angles. The absolute error is never greater than 0.00165.