Chapter 8.3, Problem 57E

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 57E

Product is $z_1{z}_2=8\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)$

Quotient is $\frac{z_1}{z_2}=2\left(\cos {90}^{\circ }+i\sin {90}^{\circ }\right)$

Explanation of Solution

Given information: Complex numbers are $z_1=4\left(\cos {120}^{\circ }+i\sin {120}^{\circ }\right)$ and $z_2=2\left(\cos {30}^{\circ }+i\sin {30}^{\circ }\right)$

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

  $\begin{array}{l}{z}_1=4\left(\cos {120}^{\circ }+i\sin {120}^{\circ }\right)\\ z_2=2\left(\cos {30}^{\circ }+i\sin {30}^{\circ }\right)\end{array}$

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

  $\begin{array}{l}{z}_1{z}_2=4\cdot 2\left(\cos \left({120}^{\circ }+{30}^{\circ }\right)+i\sin \left({120}^{\circ }+{30}^{\circ }\right)\right)\\ z_1{z}_2=8\left(\cos \left({150}^{\circ }\right)+i\sin \left({150}^{\circ }\right)\right)\\ z_1{z}_2=8\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)\end{array}$

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

  $\begin{array}{l}\frac{z_1}{z_2}=\frac{4}{2}\left(\cos \left({120}^{\circ }-{30}^{\circ }\right)+i\sin \left({120}^{\circ }-{30}^{\circ }\right)\right)\\ \frac{z_1}{z_2}=2\left(\cos \left({90}^{\circ }\right)+i\sin \left({90}^{\circ }\right)\right)\\ \frac{z_1}{z_2}=2\left(\cos {90}^{\circ }+i\sin {90}^{\circ }\right)\end{array}$

Conclusion:

Hence, Product is $z_1{z}_2=8\left(\cos {150}^{\circ }+i\sin {150}^{\circ }\right)$

Quotient is $\frac{z_1}{z_2}=2\left(\cos {90}^{\circ }+i\sin {90}^{\circ }\right)$