Answer:
z = 8cis(150)
Step-by-step explanation:
Gauss Plane
The Gauss Plane (otherwise known as the complex plane) is a cartesian-like plane that is made up by complex numbers. The x-coordinate of a point on the Gauss plane represents the real part of an imaginary number while the y-coordinate represents the coefficient to the imaginary part.
If given some point on the Gauss plane whose coordinates are (x, y), then it represents the imaginary number z = x + yi.
Polar Form
The polar form is a way to express the placement of a complex number on the Gauss Plane (otherwise known as the Imaginary Plane).
A complex number with coordinates (x, y) in polar form is as so:
[tex]z = Rcis{\theta}\\cis(\theta) = \cos\theta + i\sin\theta\\\\\text{Where:}\\R = |x + yi|\\\theta = \tan^{-1}\left(\frac yx}\right)[/tex]
In other words, R is the distance between the point (x, y) and (0, 0) on the gauss plane. Theta is the angle formed by the counterclockwise rotation of the radius R around (0, 0) starting from the positive side of the x-axis.
Breaking Down the Problem
We're given the complex number [tex]z = -4\sqrt3 + 4i[/tex] .
The coordinates of this number on the gauss plane are (-4sqrt3, 4). Thus, we can use these coordinates to calculate R and Theta per polar form:
[tex]$\begin{align*}R &= |x+yi|\\&=\sqrt{x^2 + y^2}\\&=\sqrt{(-4\sqrt3)^2 + 4^2}\\&=\sqrt{16\times3+16}\\&=\sqrt{64}\\&=8\\\\\theta&=\tan^{-1}\left(\frac xy\right)\\&=\tan^{-1}\left(\frac4{-4\sqrt3}\right)\\&=\tan^{-1}\left(-\frac1{\sqrt3}\right)\\&=-30 + 180\\&=150^{\circ}\end{align}$[/tex]
The polar form of z = -4sqrt3 + 4i is z = 8cis(150).
