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 $\sqrt{-1}$?

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)
