Another way to express $\text{cis } 75^\circ + \text{cis } 83^\circ + \text{cis } 91^\circ + \dots + \text{cis } 147^\circ$ The number
$$\text{cis } 75^\circ + \text{cis } 83^\circ + \text{cis } 91^\circ + \dots + \text{cis } 147^\circ$$ is expressed in the form $r \, \text{cis } \theta$, where $0 \le \theta < 360^\circ$. Find $\theta$ in degrees.

Hint: $\text{cis}\ \theta=\cos \theta+i \sin \theta$

Edit:
I simplified the expression down to$$\frac{\text{cis} \ 75^\circ \sin 40^\circ \ \text{cis} \ 40^\circ }{\sin 4^\circ \ \text{cis} \ 4^\circ}.$$
What should I do now?

Hint: 
$e^{i(75\pi/180)}+e^{i(83\pi/180)}+e^{i(91\pi/180)}+\cdots +e^{i(147\pi/180)}$
G.P series with common ratio $r=e^{i\frac{8\pi}{180}}$
Sum of GP series $S_n = \frac{1-r^n}{1-r}$
