Chapter 8, Problem 32RE

a.

To determine

Graph the complex number in the complex plane.

  $1+\sqrt{3i}$

Answer to Problem 32RE

The graph of complex number in the complex plane $1+\sqrt{3i}$ is,

  

Explanation of Solution

Given:

  $1+\sqrt{3i}$

Concept Used:

Desmos Graph.

According to the given details,

  $z=1+\sqrt{3i}$

  

Conclusion:

Hence, graph is

  

b.

To determine

The modulus and argument.

  $1+\sqrt{3i}$

Answer to Problem 32RE

The modulus and argument of $1+\sqrt{3i}$ is $r=2,\theta =\arctan \frac{\pi }{3}$

Explanation of Solution

Given:

  $1+\sqrt{3i}$

Concept Used:

Modulus and argument: $r=\sqrt{x^2+y^2}$

Polar form: $z=r\left(\cos \theta +i\sin \theta \right)$

Calculation:

The given complex number is $z=1+\sqrt{3i}$ ,

So, write it in the polar form,

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

Now evaluate $r$ and $\theta$ , as know that the formula for these terms,

  $r=\sqrt{x^2+y^2}$

So, from the equation,

  $\begin{array}{l}x=1\\ y=\sqrt{3}\end{array}$

Substituting these values in the above formula,

  $\begin{array}{l}r=\sqrt{1^2+{\sqrt{3}}^2}\\ r=\sqrt{1+3}\\ r=2\end{array}$

Now evaluate, $\theta$

  $\begin{array}{l}\theta =\arctan \left(\frac{\sqrt{3}}{1}\right)\\ \theta =\arctan \left(\sqrt{3}\right)\\ \theta =\frac{\pi }{3}\end{array}$

Conclusion:

Hence the modulus and argument of $1+\sqrt{3i}$ is $r=2,\theta =\frac{\pi }{3}$ .

c.

To determine

The polar form of $1+\sqrt{3i}$ .

Answer to Problem 32RE

The polar form of $1+\sqrt{3i}$ is $2\left(\cos \frac{\pi }{3}+\cos \frac{\pi }{3}\right)$

Explanation of Solution

Given:

  $1+\sqrt{3i}$

Concept Used:

Polar form: $z=r\left(\cos \theta +i\sin \theta \right)$

Calculation:

  $z=1+\sqrt{3i}$

In polar form,

  $\begin{array}{l}z=r\left(\cos \theta +i\sin \theta \right)\\ r=2\\ \theta =\frac{\pi }{3}\end{array}$

So now write these calculated values in standard equation of polar form,

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

Conclusion:

Hence, the polar form of $1+\sqrt{3i}$ is $z=2\left(\cos \frac{\pi }{3}+\cos \frac{\pi }{3}\right)$