Chapter 9.7, Problem 11E

To determine

To find the quotient and express the result in rectangular form.

Answer to Problem 11E

The value of quotient is $3\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$ and the rectangular form of the quotient is $3i$

Explanation of Solution

Given information:

The equation is

  $6\left(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}\right)÷2\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$

Calculation:

The quotient can be obtained as

  $6\left(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}\right)÷2\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$

First find the modulus and amplitude of the quotient.

  $\begin{array}{l}r=\frac{r_1}{r_2}\left[r=\text{modulus}\right]\\ r\text{ =}\frac{6}{2}\left[\text{since, }{r}_1=6\text{ and }{r}_2=2\right]\\ r=3\left[\text{divide numerator and denominator by 2}\right]\end{array}$

And

  $\begin{array}{l}\theta ={\theta }_1-{\theta }_2\left[\theta =\text{amplitude}\right]\\ \theta \text{ =}\frac{3\pi }{4}-\frac{\pi }{4}\left[\text{since, }{\theta }_1=\frac{3\pi }{4}\text{ and }{\theta }_2=\frac{\pi }{4}\right]\\ \theta \text{ =}\frac{3\pi -\pi }{4}\\ \theta =\frac{2\pi }{4}\\ \theta =\frac{\pi }{2}\left[\text{divide numerator and denominator by 2}\right]\end{array}$

The quotient is $3\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$

Now find the rectangular form of the quotient.

  $\begin{array}{l}3\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)=3\left(0+i\left(1\right)\right)\left[\text{since, }\cos \frac{\pi }{2}=0\text{ and }\sin \frac{\pi }{2}=1\right]\\ \text{ =3}\left(i\right)\\ \text{ =}3i\end{array}$

The rectangular form of the quotient is $3i$

Hence,

The value of quotient is $3\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$ and the rectangular form of the quotient is $3i$