TITLE: $e^{i\theta}$ versus $\cos\theta+i\sin\theta$
QUESTION [8 upvotes]: 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 $\operatorname{arg}(uv)=\operatorname{arg}(u)+\operatorname{arg}(v)$.

REPLY [4 votes]: Developing products of several terms is painful in the $\text{cis}$ notation as the number of terms grows exponentially.
Compare
$$e^{ia}e^{ib}e^{ic}=e^{i(a+b+c)}$$
to 
$$(\cos(a)+i\sin(a))(\cos(b)+i\sin(b))(\cos(c)+i\sin(c))=\\
\cos(a)\cos(b)\cos(c)-\sin(a)\sin(b)\cos(c)-\sin(a)\sin(c)\cos(b)-\sin(b)\sin(c)\cos(a)+\\i\sin(a)\cos(b)\cos(c)+i\sin(b)\cos(a)\cos(c)+i\sin(c)\cos(a)\cos(b)-i\sin(a)\sin(b)\sin(c)=\\
\cos(a+b+c)+i\sin(a+b+c).$$

Now try by hand
$$\frac{e^{ia}e^{ib}}{e^{ic}e^{id}}=e^{i(a+b-c-d)}$$and let me know.