Question

w=5(cos(pi/12)+i sin(pi/12) is one of six roots of complex number z. what are the other 5 roots (in trigometric form) and z?

Answer 1

Any \(n\)-th root of a complex number \(\large z=re^{i\theta}\) will have the form
\[\large z^{1/n}=r^{1/n}\left(\cos\left(\theta+\frac{2k\pi}{n}\right)+i\sin\left(\theta+\frac{2k\pi}{n}\right)\right)\]
where \(k=0,1,2,\cdots,n-1\).

You're given that one such root is \(\large w_1=5e^{i\pi/12}\), which means the other 5 sixth roots, denoted \(w_2,\cdots w_6\), are
\[\large \begin{align*}w_2&=5\left(\cos\left(\frac{\pi}{12}+\frac{2\pi}{6}\right)+i\sin\left(\frac{\pi}{12}+\frac{2\pi}{6}\right)\right)\\
w_3&=5\left(\cos\left(\frac{\pi}{12}+\frac{4\pi}{6}\right)+i\sin\left(\frac{\pi}{12}+\frac{4\pi}{6}\right)\right)\\
&\vdots\\
w_6&=5\left(\cos\left(\frac{\pi}{12}+\frac{10\pi}{6}\right)+i\sin\left(\frac{\pi}{12}+\frac{10\pi}{6}\right)\right)
\end{align*}\]
\(z\) will be any of these raised to the sixth power.