Geometric Interpretation of Trigonometric Ratios

Question

Is there a "good" geometric interpretation of trigonometric ratios for complex values? For example, we know that

$$\cos(z)=\frac{e^{iz} + e^{-iz}}{2}$$ for all complex $z$ but is there a way to interpret this geometrically?

Answer 1

Consider the unit circle in the complex plane. Then $e^{iz}$ with real $z$ covers all the points on this circle. And which turns counter clock wise. $e^{-iz}$ is the circle clock wise. Therefore with $e^{i0}=1$ the sign of the imaginary part of $e^{iz}$ and $e^{-iz}$ are opposite. The real parts are the same.

Thus $$\text{Re}(e^{iz}) = \text{Re}(e^{-iz}) $$ and $$\text{Im}(e^{iz}) = -\text{Im}(e^{-iz}) $$

$\cos z$ is defined as the real part of $e^{iz}$ which can be computed as $$\cos z= \text{Re}(e^{iz}) = \frac{e^{iz} + e^{-iz}}{2}$$ If $z$ has an imaginary part this will change the radius of your circle. With $$z=x+iy$$ $$\cos (z=x+iy) =\frac{e^{ix}e^{-y}+e^{-ix}e^{y}}{2}$$