Intuition for trigonometric function of complex number

Question

How can I intuitively understand trigonometric functions on a complex angle like what does $\cos(\theta)$ even mean when $\theta$ is complex. How can I represent this as I would have done showing a circle of radius $R$ whose any point can be represented as $P (R \cos(\theta), R \sin(\theta)) $ if $\theta$ would have been real.

Answer 1

We can refer to the expression obtained by Euler's formula

$$\cos \theta=\frac{e^{i\theta}+e^{-i\theta}}{2}$$

but of course we can't visualize that function on the complex plane in a easy way such as for the circle in the real case.

What we can do is to plot a surface for $\operatorname{Re}(\cos \theta)$, $\operatorname{Im}(\cos \theta)$ or $|\cos \theta|$ to obtain the following plots.