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 a value of $\theta$ in degrees.
I have no idea on how to deal with sequences in any way shape or form, so I am totally lost on this problem! Please help!
$$\cos75^{\circ}+\cos83^{\circ}+...+\cos147^{\circ}=$$ $$=\frac{2\sin4^{\circ}\cos75^{\circ}+2\sin4^{\circ}\cos83^{\circ}+...+2\sin4^{\circ}\cos147^{\circ}}{2\sin4^{\circ}}=$$ $$=\frac{\sin79^{\circ}-\sin71^{\circ}+\sin87^{\circ}-\sin79^{\circ}+...+\sin151^{\circ}-\sin143^{\circ}}{2\sin4^{\circ}}=$$ $$=\frac{\sin151^{\circ}-\sin71^{\circ}}{2\sin4^{\circ}}=\frac{\sin40^{\circ}\cos111^{\circ}}{\sin4^{\circ}}.$$ $$\sin75^{\circ}+\sin83^{\circ}+...+\sin147^{\circ}=$$ $$=\frac{2\sin4^{\circ}\sin75^{\circ}+2\sin4^{\circ}\sin83^{\circ}+...+2\sin4^{\circ}\sin147^{\circ}}{2\sin4^{\circ}}=$$ $$=\frac{\cos71^{\circ}-\cos79^{\circ}+\cos79^{\circ}-\cos87^{\circ}+...+\cos143^{\circ}-\cos151^{\circ}}{2\sin4^{\circ}}=$$ $$=\frac{\cos71^{\circ}-\cos151^{\circ}}{2\sin4^{\circ}}=\frac{\sin40^{\circ}\sin111^{\circ}}{\sin4^{\circ}}.$$ Thus, $$\text{cis}75^\circ + \text{cis}83^\circ + \text{cis}91^\circ + \dots + \text{cis}147^\circ=\frac{\sin40^{\circ}}{\sin4^{\circ}}\text{cis}111^{\circ}.$$ Id est, $\theta=111^{\circ}$ and we are done!