Chapter 8.3, Problem 59E

To determine

To calculate: To find the product and the quotient of the given complex numbers and express the answer in polar form

Answer to Problem 59E

Product is $z_1{z}_2=100\left(\cos {350}^{\circ }+i\sin {350}^{\circ }\right)$

Quotient is $\frac{z_1}{z_2}=\frac{4}{25}\left(\cos {50}^{\circ }+i\sin {50}^{\circ }\right)$

Explanation of Solution

Given information: Complex numbers are $z_1=4\left(\cos {200}^{\circ }+i\sin {200}^{\circ }\right)$ and $z_2=25\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)$

Formula Used:

Complex number is a number that can be expressed in the form of $a+bi$ , where $a$ and $b$ are real numbers

  $z=a+bi$

Polar form of the complex number is given as

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

Product of two complex numbers $z_1=r_1\left(\cos {\theta }_1+i\sin {\theta }_1\right)$ and $z_2=r_2\left(\cos {\theta }_2+i\sin {\theta }_2\right)$ is given as

  $z_1{z}_2=r_1{r}_2\left(\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\right)$

Quotient of two complex numbers $z_1=r_1\left(\cos {\theta }_1+i\sin {\theta }_1\right)$ and $z_2=r_2\left(\cos {\theta }_2+i\sin {\theta }_2\right)$ is given as

  $\frac{z_1}{z_2}=\frac{r_1}{r_2}\left(\cos \left({\theta }_1-{\theta }_2\right)+i\sin \left({\theta }_1-{\theta }_2\right)\right)$

Calculation:

Complex number are given as

  $\begin{array}{l}{z}_1=4\left(\cos {200}^{\circ }+i\sin {200}^{\circ }\right)\\ z_2=25\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)\end{array}$

Product of complex number is calculated as

  $z_1{z}_2=r_1{r}_2\left(\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\right)$

Substituting the values, product is

  $\begin{array}{l}{z}_1{z}_2=4\cdot 25\left(\cos \left({200}^{\circ }+{150}^{\circ }\right)+i\sin \left({200}^{\circ }+{150}^{\circ }\right)\right)\\ z_1{z}_2=100\left(\cos \left({350}^{\circ }\right)+i\sin \left({350}^{\circ }\right)\right)\\ z_1{z}_2=100\left(\cos {350}^{\circ }+i\sin {350}^{\circ }\right)\end{array}$

Quotient of complex number is calculated as

  $\frac{z_1}{z_2}=\frac{r_1}{r_2}\left(\cos \left({\theta }_1-{\theta }_2\right)+i\sin \left({\theta }_1-{\theta }_2\right)\right)$

Substituting the values, product is

  $\begin{array}{l}\frac{z_1}{z_2}=\frac{4}{25}\left(\cos \left({200}^{\circ }-{150}^{\circ }\right)+i\sin \left({200}^{\circ }-{150}^{\circ }\right)\right)\\ \frac{z_1}{z_2}=\frac{4}{25}\left(\cos \left({50}^{\circ }\right)+i\sin \left({50}^{\circ }\right)\right)\\ \frac{z_1}{z_2}=\frac{4}{25}\left(\cos {50}^{\circ }+i\sin {50}^{\circ }\right)\end{array}$

Conclusion:

Hence, Product is $z_1{z}_2=100\left(\cos {350}^{\circ }+i\sin {350}^{\circ }\right)$

Quotient is $\frac{z_1}{z_2}=\frac{4}{25}\left(\cos {50}^{\circ }+i\sin {50}^{\circ }\right)$