Find the roots, $z_1$&$z_2$ of $z^2 - (4\cos a)z+4=0$, where $a$ is a constant and $0<a<(\pi)/2$, expressing them in polar form..

Question

$z^2$ -4cos(a)z+4=0, where 0 < a< $\pi$/2. In other complex polynomials, I had been able to find their roots by completing the square however in this case, I unsuccessfully tried, and do not know where to go from here. According to the source the roots are: 2cis(a) or 2cis(-a) OR alternatively 2cos(a)+2sin(a)i or 2cos(-a)+ 2sin(-a)i.

Answer 1

$(z-2\cos\, a)^{2}=4\cos^{2}\, a-4=-4\sin^{2}\, a$. So $z-2\cos\, a=\pm 2i \sin \, a$ and $z=2\cos\, a\pm 2i \sin \, a$.