$e^{i\theta}$ versus $\cos\theta+i\sin\theta$ I am teaching an basic university maths course, and have been thinking about the complex numbers part. Specifically, I was wondering why I should include Euler's formula in my course.
This led me to the following "big list" question, which I thought interesting.

What can you do with $e^{i\theta}$ that is much harder/impossible with $\cos\theta+i\sin\theta$?

For example, $e^{i\theta}$ makes it easy to see that $\operatorname{arg}(uv)=\operatorname{arg}(u)+\operatorname{arg}(v)$.

Complex Analysis uses a lot the logarithm. As you said with $e^{i\theta}$ you can talk more easily for the argument. The important thing is that a branch of the argument exists iff exist a branch of the logarithm. 
Therefore, exponential goes with logarithm. Also of the expressions $z^a$ are exponential forms...
