Chapter 9, Problem 50SGA

To determine

To express: the complex number in polar form.

Answer to Problem 50SGA

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

Explanation of Solution

Given information:

  $3i$

Calculation:

The coordinates $\left(x,y\right)$ obtained from the complex number is $\left(0,3\right)$ . Put the coordinates $\left(x,y\right)$ in the formula given below and evaluate the value of $r$ .

  $\begin{array}{c}r=\sqrt{x^2+y^2}\\ =\sqrt{{\left(0\right)}^2+{\left(3\right)}^2}\\ =\sqrt{9}\\ =3\end{array}$

Since we have an imaginary number and $3>0$ so, conclude that $\theta =\frac{\pi }{2}$ .

The polar form can be evaluated as shown below.

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

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