Chapter 9.7, Problem 19E

To determine

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

Answer to Problem 19E

The value of quotient is $4\left(\cos \left(-\text{5}\text{.6}\right)+i\sin \left(-\text{5}\text{.6}\right)\right)$ and the rectangular form of the quotient is $\text{3}\text{.08}-2.52i$

Explanation of Solution

Given information:

The equation is

  $4\left(\cos \left(-2\right)+i\sin \left(-2\right)\right)÷\left(\cos 3.6+i\sin 3.6\right)$

Calculation:

The quotient can be obtained as

  $4\left(\cos \left(-2\right)+i\sin \left(-2\right)\right)÷\left(\cos 3.6+i\sin 3.6\right)$

First find the modulus and amplitude of the quotient.

  $\begin{array}{l}r=\frac{r_1}{r_2}\left[r=\text{modulus}\right]\\ r\text{ =}\frac{4}{1}\left[\text{since, }{r}_1=4\text{ and }{r}_2=1\right]\\ r=4\end{array}$

And

  $\begin{array}{l}\theta ={\theta }_1-{\theta }_2\left[\theta =\text{amplitude}\right]\\ \theta \text{ =}-\text{2}-3.6\left[\text{since, }{\theta }_1=-2\text{ and }{\theta }_2=3.6\right]\\ \theta \text{ =}-\text{5}\text{.6 }\left[\theta \text{ in radian}\right]\end{array}$

The quotient is $4\left(\cos \left(-\text{5}\text{.6}\right)+i\sin \left(-\text{5}\text{.6}\right)\right)$

Now find the rectangular form of the quotient.

  $\begin{array}{l}4\left(\cos \left(-\text{5}\text{.6}\right)+i\sin \left(-\text{5}\text{.6}\right)\right)=4\left(0.77+i\left(-0.63\right)\right)\left[\text{since, }\cos \left(-5.6\right)=0.77\text{ and }\sin \left(-5.6\right)=-0.63\right]\\ =4\left(0.77-0.63i\right)\\ \text{ =3}\text{.08}-2.52i\end{array}$

The rectangular form of the quotient is $\text{3}\text{.08}-2.52i$

Hence,

The value of quotient is $4\left(\cos \left(-\text{5}\text{.6}\right)+i\sin \left(-\text{5}\text{.6}\right)\right)$ and the rectangular form of the quotient is $\text{3}\text{.08}-2.52i$