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 $\sqrt{-1}$?

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
