How to Multiply and Divide Complex Numbers in Polar Form?

When two or more complex numbers are multiplying is a fundamental operation. Compared to the adding and subtracting of complex numbers, it is a more complicated process.

When a real number is multiplied by an imaginary number, you get a “complex number,” which is \(a+ib\).

Complex number multiplication is similar to binomial multiplication utilizing the distributive property.

Related Topics

• How to Multiply and Divide Complex Numbers
• How to Add and Subtract Complex Numbers

A step-by-step guide to multiplying and dividing complex numbers in polar form

To multiply complex numbers in the polar form, follow these steps:

• When two complex numbers are multiplied in the polar form \(z_1=r_1(cos θ_1+i\sin θ_1)\) and \(z_2=r_2(cos θ_2+i\sin θ_2)\), to find out what their output is, apply the formula below: \(\color{blue}{z_1z_2=r_1 r_2[cos( θ_1 + θ_2)+i\sin ( θ_1 + θ_2)]}\)

To divide complex numbers in the polar form, follow these steps:

• In the first step, identify the components of the complex number: \(r_1\), \(r_2\), \( θ_1 \), and \( θ_2\).

• One thing to do now is to put the numbers found in Step 1 into the formula for dividing complicated numbers in the “polar” form. \(\color{blue} {\frac{z_1}{z_2}=\frac{r_1}{r_2} [cos( θ_1 – θ_2 )+i\sin ( θ_1 – θ_2)]}\)

• Make things as simple as possible.

Multiplying and Dividing Complex Numbers in Polar Form, Example 1:

Find the product of \(z_1 z_2\).

\(z_1=3 (cos(75) + i sin (75))\) and \(z_2=2 (cos(150) + i sin (150))\)

To find \(z_1 z_2\) use this formula: \(\color{blue}{z_1z_2=r_1 r_2[cos( θ_1 + θ_2)+i\sin ( θ_1 + θ_2)]}\)

\(z_1 z_2= 3\times2 [cos (75+150) + i\ sin (75+150)]\)

\(z_1 z_2\) \(= 6[cos (225) + i\ sin( 225)]\)

\(z_1 z_2\) \(=6 [cos (\frac {5π}{4}) + i\ sin (\frac {5π}{4})]\)

\(z_1 z_2\) \(=6 [-\frac{\sqrt 2}{2} + i (-\frac{\sqrt 2}{2})]\)

\(z_1 z_2\) \(=-3\sqrt {2}-3i\sqrt{2}\)

Multiplying and Dividing Complex Numbers in Polar Form, Example 2:

Find the quotient of \(\frac {z_1}{z_2}\).

\(z_1=2 (cos(210) + i sin (210))\) and \(z_2=8 (cos(30) + i sin (30))\)

To find \(\frac {z_1}{z_2}\) use this formula: \(\color{blue} {\frac{z_1}{z_2}=\frac{r_1}{r_2} [cos( θ_1 – θ_2 )+i\sin ( θ_1 – θ_2)]}\).

\(\frac{z_1}{z_2}=\frac{2}{8}[cos(210-30) + i\ sin (210-30)]\)

\(\frac{z_1}{z_2}= \frac{1}{4}[cos (180) + i\ sin (180)]\)

\(\frac{z_1}{z_2}= \frac{1}{4}[-1+0 i]\)

\(\frac{z_1}{z_2}= -\frac{1}{4}+0 i\)

\(\frac{z_1}{z_2}= -\frac{1}{4}\)

Exercises for Multiplying and Dividing Complex Numbers in Polar Form

Find each product. 

• \(\color{blue}{z_1= 2\sqrt{2}[cos (145)+i sin(145)]}\) and \(\color{blue}{z_2= 2[cos (35)+i sin(35)]}\)
• \(\color{blue}{z_1= 2 [cos (215)+i sin(215)]}\) and \(\color{blue}{z_2=8 [cos (25)+i sin(25)]}\)

Find each quotient.

• \(\color{blue}{z_1= 10[cos (145)+i sin(145)]}\) and \(\color{blue}{z_2= 5[cos (25)+i sin(25)]}\)
• \(\color{blue}{z_1= 4[cos (150)+i sin(150)]}\) and \(\color{blue}{z_2= 8[cos (90)+i sin(90)]}\)

• \(\color{blue}{z_1 z_2=-4\sqrt{2}}\)
• \(\color{blue}{z_1 z_2=-8-8\sqrt{3}i}\)
• \(\color{blue}{\frac{z_1}{z_2}=-1+\sqrt{3}i}\)
• \(\color{blue}{\frac{z_1}{z_2}=\frac{1}{4}+\frac{\sqrt3}{4}i}\)