Chapter 9, Problem 45SGA

To determine

To express: the complex number in polar form.

Answer to Problem 45SGA

  $2\left(\cos \left(\frac{2\pi }{3}\right)+i\sin \left(\frac{2\pi }{3}\right)\right)$

Explanation of Solution

Given information:

  $-1+\sqrt{3}i$

Calculation:

The coordinates $\left(x,y\right)$ obtained from the complex number is $\left(-1,\sqrt{3}\right)$ . Put the coordinates $\left(x,y\right)$ in the formula given below and evaluate the value of $r$ .

  $\begin{array}{c}r=\sqrt{x^2+y^2}\\ =\sqrt{{\left(-1\right)}^2+{\left(\sqrt{3}\right)}^2}\\ =\sqrt{4}\\ =2\end{array}$

Put the coordinates $\left(x,y\right)$ in the formula given below and evaluate $\theta$ .

  $\begin{array}{c}\theta =\arctan \left(\frac{y}{x}\right)\text{+}\pi \left[\because x<0\right]\\ =\arctan \left(\frac{\sqrt{3}}{-1}\right)\text{+}\pi \\ =-\frac{\pi }{3}\text{+}\pi \\ =\frac{2\pi }{3}\end{array}$

The polar form can be evaluated as shown below.

  $\begin{array}{c}\text{Polar form}=r\left(\cos \theta +i\sin \theta \right)\\ =2\left(\cos \left(\frac{2\pi }{3}\right)+i\sin \left(\frac{2\pi }{3}\right)\right)\left[\text{Put }r\text{ and }\theta \right]\end{array}$

Thus the polar form of the given complex number is $2\left(\cos \left(\frac{2\pi }{3}\right)+i\sin \left(\frac{2\pi }{3}\right)\right)$ .