How does the complex exponential function transform the unit circle?

I know you can write every complex number on the unit circle as $e^{i\theta} = \cos(\theta)+i\sin(\theta).$ But what does it look like when you raise $e$ to the values? You get $e^{\cos(\theta)+i\sin(\theta)} = e^{\cos(\theta)}(\cos(\sin(\theta)) + i\sin(\sin(\theta))),$ but I'm having trouble visualizing this.