Complex sine and cosine

Definition

For $z \in \mathbb{C}$, the complex sine and the complex cosine of $z$ are $$\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \qquad \cos z = \frac{e^{iz} + e^{-iz}}{2},$$ where $e^w$ is the complex exponential of $w$.

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.

Explanations

L1 human

Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$ lets us solve for $\cos \theta$ and $\sin \theta$ in terms of exponentials. These formulas make sense for every complex number, so we use them to define $\sin z$ and $\cos z$. On the imaginary axis they grow very fast: $\cos(10 i)$ is more than $10000$.