Question

analyzing roots of a complex number
consider the 4th roots of 16cos(π) + i sin(π).
the roots are located on a circle with center at the pole and radius of
the arguments of two successive roots differ by π units along the circumference of a circle.

Explanation:

Step1: Find radius of root circle

For a complex number $r[\cos\theta + i\sin\theta]$, the radius of the circle containing its $n$th roots is $\sqrt[n]{r}$. Here, $r=16$, $n=4$.
$\sqrt[4]{16} = 2$

Step2: Find argument difference

The difference between arguments of successive $n$th roots is $\frac{2\pi}{n}$. Here, $n=4$.
$\frac{2\pi}{4} = \frac{1}{2}\pi$

Answer:

1. $2$
2. $\frac{1}{2}$