Complex Numbers Midpoint of Roots of Unity

Question

A = $\sqrt{2}e^{i(\frac{7\pi}{12})}$

B = $\sqrt{2}e^{i(\frac{11\pi}{12})}$

Express the midpoint M of AB in the form $a + bi$

(a,b in simplified surd form)

I know M = (A+B)/2 but I cant find A+B in a simple enough form.

Answer 1

First of all I'd convert $A$ and $B$ to cartesian coordinates:

$A = \sqrt{2}(\cos(7\pi/12)+i \sin(7\pi/12))$

$B= \sqrt{2}(\cos(11\pi/12)+i \sin(11\pi/12))$

Then you can easily compute

$M=(A+B)/2 = \frac{\sqrt{2}}{2} (\cos(7\pi/12)+\cos(11\pi/12)+i\left[\sin(7\pi/12)+\sin(11\pi/12)\right])$