Chapter 8.3, Problem 51E

To determine

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

Answer to Problem 51E

Complex number in polar form is $z=2\left(\cos \frac{\pi }{4}+isin\frac{\pi }{4}\right)$

Explanation of Solution

Given information: Complex numbers is $z=\sqrt{2}+\sqrt{2}i$

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

  $z=\sqrt{2}+\sqrt{2}i$

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,

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

Calculating the value of $\theta$ as follows:

  $\tan \theta =\frac{b}{a}$

Substituting the values,

  $\begin{array}{l}\tan \theta =\frac{\sqrt{2}}{\sqrt{2}}\\ \tan \theta =1\\ \theta ={\tan }^{-1}1\end{array}$

Here, solution lies in first quadrant

  $\theta =\frac{\pi }{4}$

Now, substituting the values of $r=2$ and $\theta =\frac{\pi }{4}$ in equation (1) to get the complex number in polar form

  $z=2\left(\cos \frac{\pi }{4}+isin\frac{\pi }{4}\right)$

Conclusion:

Hence, complex number in polar form is $z=2\left(\cos \frac{\pi }{4}+isin\frac{\pi }{4}\right)$