If you use the identity z^n = cos(nθ)+isin(nθ) the equation becomes
cos(28θ)+isin(28θ)cos(8θ)isin(8θ)1=0
Separating this into real and imaginary components gives the system:
cos(28θ)cos(8θ)1=0
sin(28θ)sin(8θ)=0
Note that each of these is periodic with period 360/4 = 90 degrees (4 being the GCD of 8 and 28.)
This could be used along with a lot of identities to reduce and solve analytically, but I didn't feel like it. So I just made a table to get the solutions of each on the interval [0,90)
The cosines equation has 8 solutions and the sine has 12. The two in common are 15 and 75.
Add the period to give all the solutions sought:
θ={15,75,105,165,195,255,285,345}

Posted by Jer
on 20151126 09:51:41 