# How can I visualize the interaction of the imaginary parts of the cosine/sine functions?

So I've been trying to get a good and intuitive feel for the extension of the sine and cosine functions into complex numbers (i.e. $\cos(z)$ where $z=a+bi$), and to do so I've naturally been looking at their definition using exponential functions

$$\cos(x)=\frac{e^{ix}+e^{i(-x)}}{2}$$ $$\sin(x)=\frac{e^{ix}-e^{i(-x)}}{2}$$

If we let $f(x)=e^{ix}$ and $g(x)=e^{i(-x)}$ , is there any way to imagine $f(x)$ or $g(x)$ on a purely real plane (or at least their influences that lead them to give "real" $\sin(x)$ and $\cos(x)$ graphs)? I understand that they form a circle when graphed in the complex plane, and to my understanding the cosine function is the real part of $e^{ix}$ while the sine function is the imaginary part when graphed against time, but I'm wondering if there is a more intuitive and less quantitative way to understand the relationship between complex exponential and trigonometric functions.

• With gusto!${}$ – user98602 Jul 20 '15 at 5:38
• I would start drawing $f(x)=e^{ix}$ as a 3D curve with axis Z related to the imaginary part of the image. – Masacroso Jul 20 '15 at 8:42