Chapter 9, Problem 30RE

To determine

Towrite:the given complex number in the standard form and plot in the complex plane.

Answer to Problem 30RE

  $z=\frac{3}{2}+\frac{3\sqrt{3}}{2}i$ .

Explanation of Solution

Given:

The givencomplex number is $3\left(\cos {60}^{\circ }+i\sin {60}^{\circ }\right)$ .

Calculation:

The standard form of complex plane is $z=a+ib=r\left(\cos \theta +i\sin \theta \right)$

As the given complex number is $3\left(\cos {60}^{\circ }+i\sin {60}^{\circ }\right)$ . Therefore,

  $\begin{array}{l}a+ib=3\left(\cos {60}^{\circ }+i\sin {60}^{\circ }\right)\\ \Rightarrow a+ib=\left(3\cos {60}^{\circ }+3i\sin {60}^{\circ }\right)\\ a+ib=3\left(\frac{1}{2}\right)+i3\left(\frac{\sqrt{3}}{2}\right)\\ \therefore a+ib=\frac{3}{2}+\frac{3\sqrt{3}}{2}i\end{array}$ .

Now, draw the graph of the complex plane $z=3\left(\cos {60}^{\circ }+i\sin {60}^{\circ }\right)$ .

Hence, the given complex number in the standard form $a+ib=\frac{3}{2}+\frac{3\sqrt{3}}{2}i$