Question

57Convert4cis 3 to rectangular form.O A.-2 – 2/31OB. -2 + 2/31O C. 2 - 2/31OD. 2 + 2/31

Answer 1

The question

$\sqrt[]{3}+i$

is written in the standard rectangular form:

$a+bi$

To write in polar form, we must write it in the format

$\begin{gathered} r(\cos \theta+i\sin \theta) \\ or \\ rcis\theta \end{gathered}$

To find r, we can use the formula

$\begin{gathered} r=\sqrt[]{a^2+b^2} \\ \text{where} \\ a=\sqrt[]{3} \\ b=1 \end{gathered}$

Solving, we have

$\begin{gathered} r=\sqrt[]{(\sqrt[]{3})^2+1^2} \\ r=\sqrt[]{3+1}=\sqrt[]{4} \\ r=2 \end{gathered}$

To find θ, we use

$\theta=\tan ^{-1}\frac{b}{a}$

Substituting the values, we have

$\begin{gathered} \theta=\tan ^{-1}\frac{1}{\sqrt[]{3}} \\ \theta=30^{\circ} \end{gathered}$

In polar form,

$\theta=\frac{\pi}{6}$

Note that since a and b are positive, the angle is in the first quadrant. Hence, we use the angle as is.

Therefore, we have the answer to be

$2\text{cis}\frac{\pi}{6}$

OPTION A is correct.