Chapter 15.4, Problem 58E

To determine

To find:the value of the given function and express the result in rectangular form.

Answer to Problem 58E

  $1+\sqrt{3}i$ .

Explanation of Solution

Given:

The given function is $2\sqrt{2}\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)÷\sqrt{2}\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)$ .

Calculation:

As the given function is $\frac{2\sqrt{2}\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)}{\sqrt{2}\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)}$ .

divide the magnitudes and subtract the angles so that to find the value of the given function. Then

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

To find the rectangular form we plug in the values of the sine and cosine.

  $\begin{array}{l}2\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)=2\left(\cos \frac{1}{2}+i\frac{\sqrt{3}}{2}\right)\\ =1+\sqrt{3}i\end{array}$

Hence, the value of the given functionin rectangular form is $1+\sqrt{3}i$ .