Question

Let z1=a(cos pi/4 + i sin pi/4) and z2=b(cos pi/3 + i sin pi/3).

Express (z1/z2)^3 in the form z=x+yi.

Answer 1

\[e^{\theta i}=cis(\theta) = \cos(\theta) + i \sin(\theta)\]
so, z1 = a*e^(i*pi/4) and z2 = b*e^(i*pi/3)
z1/z2 = (a/b)*(e^(-i*pi/12))
\[(z1/z2)^3 = (a^3/b^3)*(e^{\frac{-i \pi}{4}}) = \frac{a^3}{b^3} \cos( -\pi/4)+i \frac{a^3}{b^3} \sin(-\pi/4) = \frac{a^3 \sqrt{2}}{b^3 2}+i \frac{a^3 \sqrt{2}}{b^3 2} = z\]