$$\begin{aligned} \cos x &= \frac{e^{ix} + e^{-ix}}{2} \\ \sin x &= \frac{e^{ix} - e^{-ix}}{2i} \\ \end{aligned}$$

This works whether $x$ is $\theta \in \mathbb{R}$ or $z \in \mathbb{C}$. Do you see why ?

Furthermore, it's not difficult to show their relationships with hyperbolic functions:

$$\begin{aligned} \sin z &= \sin(x + iy) = \sin x\cosh y + i\cos x\sinh y \\ \cos z &= \cos(x + iy) = \cos x\cosh y\, – i\sin x\sinh y \\ \end{aligned}$$

Why would this be useful ? It's particularly handy when there is the need to evaluate some form of conjugate or absolute value, where it's helpful to break down the function into real and imaginary parts.

Did I mention already ? Euler, the truly incredible.

# posted by rot13(Unafba Pune) @ 8:20 AM