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

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 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 \(\color{blue}{z_2= 8[cos (90)+i sin(90)]}\)

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