Exercise 1.4 — Question 9

Topic: Operations with Complex Numbers in Polar Form

This question applies the polar (trigonometric) representation of complex numbers and the rules for multiplication, division, and powers in polar form.

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Key Concepts

Polar Coordinate System

In the polar coordinate system, a point $P$ in the plane is located by:

• $r$ — the radial distance from the origin (pole) to $P$, where $r\ge 0$
• $\theta$ — the polar angle (argument) measured counter-clockwise from the positive $x$-axis

So $P=(r,\theta )$ in polar form.

Polar Representation of a Complex Number

For $z=a+ib$, the polar (trigonometric) form is:

$z=r(\cos \theta +i\sin \theta )$

where:

• $r=|z|=\sqrt{a^2+b^2}$ is the modulus
• $\theta =\arg (z)={\tan }^{-1}\left(\frac{b}{a}\right)$ is the argument (adjusted for the correct quadrant)
• $a=r\cos \theta$, $b=r\sin \theta$

Operations in Polar Form

Multiplication: $z_1{z}_2=r_1{r}_2[\cos ({\theta }_1+{\theta }_2)+i\sin ({\theta }_1+{\theta }_2)]$

Division: $\frac{z_1}{z_2}=\frac{r_1}{r_2}[\cos ({\theta }_1-{\theta }_2)+i\sin ({\theta }_1-{\theta }_2)],z_2\ne 0$

De Moivre's Theorem: $z^n=r^n(\cos n\theta +i\sin n\theta )$

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Worked Example (Q-9 Type)

Express the following in polar form and perform the indicated operation.

Example: Let $z_1=1+i$ and $z_2=\sqrt{3}-i$. Find $z_1{z}_2$ in polar form.

Step 1 — Convert $z_1=1+i$ to polar form: $r_1=\sqrt{1^2+1^2}=\sqrt{2},{\theta }_1={\tan }^{-1}\left(\frac{1}{1}\right)=45°$ $z_1=\sqrt{2}(\cos 45°+i\sin 45°)$

Step 2 — Convert $z_2=\sqrt{3}-i$ to polar form: $r_2=\sqrt{(\sqrt{3}{)}^2+(-1{)}^2}=\sqrt{3+1}=2$ ${\theta }_2={\tan }^{-1}\left(\frac{-1}{\sqrt{3}}\right)=-30°(\text{4th quadrant})$ $z_2=2(\cos (-30°)+i\sin (-30°))$

Step 3 — Multiply using the polar multiplication rule: $z_1{z}_2=\sqrt{2}\cdot 2[\cos (45°+(-30°))+i\sin (45°+(-30°))]$ $=2\sqrt{2}(\cos 15°+i\sin 15°)$

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Summary Table