Equation visualization of complex sinusoid function
The exponent function is directly related to the sine and cosine via Euler equation:

$$e^{i \theta} = \cos(\theta) + i \sin(\theta)$$

For example for function $z=e^{i \pi t}$:

$$z=\cos(\pi t) + i \sin(\pi t) $$



You could plot it in 3D, where x axis is the time, and y &amp; z axes are real and imaginary part of your complex signal, respectively. In 3D that is going to be a helix:



But if you look at projections onto the y=0 and z=0 planes, then you will notice sine:



and cosine waves:



On the other hand, if you look at the x=0 projection, then you will observe a (unit) circle:



You can find the applet with this visualization in Wolfram Demonstration Project.