Complex sine and cosine

Remarks

For real $z$, these are the usual sine and cosine, by Euler's formula. The functions $\sin$ and $\cos$ are entire, with $\sin' = \cos$ and $\cos' = -\sin$. The identity $\sin^2 z + \cos^2 z = 1$ holds for each $z \in \mathbb{C}$. They are not bounded on $\mathbb{C}$. For real $y$, $\cos(iy) = \cosh y = \frac{e^y + e^{-y}}{2}$. The power series of the cosine is the same as in the real case.