Chapter 9, Problem 29RE

To determine

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

Answer to Problem 29RE

  $z=-\sqrt{3}+i$ .

Explanation of Solution

Given:

The givencomplex number is $2\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)$ .

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 $2\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)$ . Therefore,

  $\begin{array}{l}a+ib=2\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)\\ \Rightarrow a+ib=2\left(\cos {\left(180-30\right)}^{\circ }+i\sin {\left(180-30\right)}^{\circ }\right)\\ \Rightarrow a+ib=\left(-2\cos {30}^{\circ }+2i\sin {30}^{\circ }\right)\\ a+ib=-2\left(\frac{\sqrt{3}}{2}\right)+i2\left(\frac{1}{2}\right)\\ \therefore a+ib=-\sqrt{3}+i\end{array}$ .

Now, draw the graph of the complex plane $z=2\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)$ .

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