How to calculate the sine manually, without any rules, calculator or anything else? 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 $\sqrt{-1}$?
 A: Use the old-fashioned method: draw a really big circle, add the angle you wish to calculate, and measure.
A: 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.
A: You changed very quickly in a comment to MrFatzo's answer to "how do computers calculate $\sin(x)$?", so I'm going to infer that what you're actually trying to ask is:

How does one calculate sines from scratch, without taking anyone's word for the correctness of tables or other magic values that go into the calculation?

I'm aware of two methods:


*

*The ancients reckoned sines in degrees rather than radians. They created tables of sine values (actually chord values, in really ancient times, but that more or less amounts to the same problem) by starting with $\sin(0^\circ)=0$, $\sin(90^\circ)=1$ and then using known formulas for $\sin(v/2)$ to find sines of progressively smaller angles than $90^\circ$, and then formulas for $\sin(v+u)$ to find sines of sums of these smaller angles. That way they could eventually fill out their entire table.
In this method calculating a single sine from scratch is not really anything you do -- it's not very much less work than creating the entire table, which is to say: years and years of painstaking manual calculations.
See How to evaluate trigonometric functions by pen and paper? for a bit more detail.

*In more modern times -- that is, roughly after the development of calculus -- we prefer our sines in radians. Then the gold standard for what the value of a sine should be is the power series:
$$ \sin x = x - \frac16 x^3 + \frac1{120} x^5 - \cdots + \frac{(-1)^n}{(2n+1)!} x^{2n+1} + \cdots $$
This series converges quite fast when $x$ is not larger than a handful of radians, and it is simple to estimate the convergence as you go along (once $2n>x$, the limit will be strictly between any two successive partial sums), so that lets you compute single sines from scratch to any precision you desire.
The power series is still kind of slow even for computers, if you want to compute millions of sines. So in practice computers and calculators use various combinations of clever interpolation methods and tables that are built into the hardware. The tables themselves were ultimately constructed using the power series methods.
A: There are many ways in which trig functions are calculated by computers, including rather inaccurate ones (for example the fsin method of Intel processors is notorious). A nice overview of implementations of many  functions can be found here as part of the GNU MPFR library.
A: I'm not sure what can you do "manually", but maybe try using a taylor approximation?
For example, you can calculate $x-\frac{x^3}{6}$ 
A: As for how computer actually evaluate sin(x) and other trig / transcendental functions, rather than using the Taylor series, which can converge rather slowly at times, the method usually used is a Chebyshev Polynomial.  It should be noted that the whooshing sound you can hear is the mathematics on that page going clean over my head. ;)
That said, you normally extract a relatively small number of coefficients, and use them in a polynomial expansion that gets reasonable accuracy, albeit with a non-zero error term.  This page shows the numbers involved in evaluating Sin(x)
A: Well you can use the taylor series or you can use an approximation which I am suggesting, but it will be only useful if you know the value of sin of some nearby angles
