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.
I tried to use this site for geometric sequences, How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?.
Hint:
Let $\frac{75\pi}{180}=\alpha$. Since $\text{cis} \theta=e^{i \theta}$. Therefore the given series can be written as $$e^{i \alpha}+e^{i \left(\alpha+\frac{8\pi}{180}\right)}+e^{i \left(\alpha+\frac{16\pi}{180}\right)}+ \dotsb + e^{i \left(\alpha+\frac{72\pi}{180}\right)}.$$ Let $\frac{8\pi}{180}=\beta$. Then we can write this as: $$e^{i \alpha}\left[1+e^{i \beta}+e^{i2\beta}+ \dotsb + e^{i 9\beta}\right].$$ This is a geometric series with common ratio $e^{i\beta}$.
\begin{align} \text{cis }75^\circ + \text{cis }83^\circ + \text{cis }91^\circ + \dots + \text{cis }147^\circ &= \text{cis }75^\circ \left( 1 + \text{cis }8^\circ + \text{cis }16^\circ + \dots + \text{cis }72^\circ \right)\\ &= \text{cis }75^\circ \left( 1 + \text{cis }8^\circ + (\text{cis }8^\circ)^2 + \dots + (\text{cis }8^\circ)^9 \right)\\ &= \text{cis }75^\circ \dfrac{(\text{cis }8^\circ)^{10}-1}{\text{cis }8^\circ-1}\\ &= \text{cis }75^\circ \dfrac{\text{cis }80^\circ-1}{\text{cis }8^\circ-1}\\ \end{align}
