Chapter 8.3, Problem 30E

To determine

To calculate: To write the complex number in polar form with argument $\theta$ between $0$ and $2\pi$

Answer to Problem 30E

Complex number in polar form is $z=2\left(\cos \frac{\pi }{3}+isin\frac{\pi }{3}\right)$

Explanation of Solution

Given information: Complex numbers is $z=1+\sqrt{3}i$

Formula Used:

Complex number is a number that can be expressed in the form of $a+bi$ , where $a$ and $b$ are real numbers

  $z=a+bi$

Polar form of the complex number is given as

  $z=r\left(\cos \theta +isin\theta \right)$

Where,

  $r=\left|z\right|=\sqrt{a^2+b^2}$

  $\tan \theta =\frac{b}{a}$

Calculation:

Complex number is given as

  $z=1+\sqrt{3}i$

Polar form of the complex number is given as

  $z=r\left(\cos \theta +isin\theta \right)----(1)$

Calculating the value of $r$ as follows:

  $r=\left|z\right|=\sqrt{a^2+b^2}$

Substituting the values,

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

Calculating the value of $\theta$ as follows:

  $\tan \theta =\frac{b}{a}$

Substituting the values,

  $\begin{array}{l}\tan \theta =\frac{\sqrt{3}}{1}\\ \tan \theta =\sqrt{3}\\ \theta ={\tan }^{-1}\sqrt{3}\end{array}$

Here, solution lies in first quadrant

  $\theta =\frac{\pi }{3}$

Now, substituting the values of $r=2$ and $\theta =\frac{\pi }{3}$ in equation (1) to get the complex number in polar form

  $z=2\left(\cos \frac{\pi }{3}+isin\frac{\pi }{3}\right)$

Conclusion:

Hence, complex number in polar form is $z=2\left(\cos \frac{\pi }{3}+isin\frac{\pi }{3}\right)$