Finding the exponential form of $e^{17\pi i/60} + e^{27\pi i/60} + e^{37\pi i /60} + e^{47 \pi i /60} + e^{57 \pi i/60}$ 
Find the exponential form of the complex number
$$ e^{17\pi i/60} + e^{27\pi i/60} + e^{37\pi i /60} + e^{47 \pi i /60} + e^{57 \pi i/60}$$ with proof.

I know that I have to combine some of the exponents and simplify them, but I am having some trouble with it. For example, $e^{17 \pi i/60} + e^{57 \pi i/60} = e^{17 \pi i/60} (1+e^{40})$ but I'm having trouble connecting it with the other terms.
Can anybody help? Thanks!
 A: You have\begin{align}e^{17\pi i/60}+e^{27\pi i/60}+\cdots+e^{57\pi i/60}&=e^{17\pi i/60}\left(1+e^{\pi i/6}+\cdots+e^{4\pi i/6}\right)\\&=e^{17\pi i/60}\frac{1-e^{5\pi i/6}}{1-e^{\pi i/6}}\\&=e^{17\pi i/60}\left(\frac12\left(2+\sqrt3\right)+\frac12\left(3+2\sqrt3\right)i\right)\\&=\left(2+\sqrt3\right)e^{17\pi i/60}\left(\frac12+\frac{\sqrt3}2i\right)\\&=\left(2+\sqrt3\right)e^{17\pi i/60}e^{\pi i/3}\\&=\left(2+\sqrt3\right)e^{37\pi i/60}.\end{align}
A: My sol.
We first look at $e^{17 \pi i/60} + e^{57 \pi i/60}$ and represent it in polar form.
We note that $$e^{17 \pi i/60} = \cos \frac{17\pi}{60} + i\sin\frac{17\pi}{60}$$ and $$e^{57 \pi i/60} = \cos \frac{57\pi}{60} + i\sin\frac{57\pi}{60}.$$
Therefore, by adding the cosines and sines we get $$e^{17 \pi i/60} + e^{57 \pi i/60} = 2\cos\frac{37\pi}{60}\cos\frac{\pi}{3} + i(2\sin\frac{37\pi}{60}\cos\frac{\pi}{3})$$ which simplifies to $\text{cis} \frac{37\pi}{60}$ after evaluating the cosines and sines of common angles.
We can simplify $e^{27 \pi i/60} + e^{47 \pi i/60}$ in the same way.
We can first note that $$e^{27 \pi i/60} = \cos \frac{27\pi}{60} + i\sin\frac{27\pi}{60}$$ and $$e^{47 \pi i/60} = \cos \frac{47\pi}{60} + i\sin\frac{47\pi}{60}.$$ Therefore after collecting like terms and simplifying cosines and sines that we know on the top of our heads, we get that $$e^{27 \pi i/60} + e^{47 \pi i/60} = \sqrt{3}(\text{cis}\frac{37\pi}{60}).$$
Finally $e^{37\pi i /60}$ can be simply simplified to $$\text{cis} \frac{37 \pi}{60}.$$
Adding the individual terms, we finalize to $$(\sqrt{3}+2) \text{cis} \frac{37 \pi}{60}.$$
Therefore, $$e^{17\pi i/60} + e^{27\pi i/60} + e^{37\pi i /60} + e^{47 \pi i /60} + e^{57 \pi i/60} = \boxed{(\sqrt{3}+2) e^{\frac{37\pi i}{60}}}.$$
