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Polar Form of a Complex Number

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Polar Form of a Complex Number

Study Guide

Key Definition

The polar form of a complex number is expressed as $z = r(\cos(\theta) + i\sin(\theta))$

Important Notes

In polar form, $r$ is the distance from the origin.
The angle $\theta$ is measured from the positive x-axis.
Convert from Cartesian to polar using $r = \sqrt{x^2 + y^2}$ and $\theta = \mathrm{Arg}(x + yi) = \tan^{-1}(\frac{y}{x})$ (with quadrant adjustment).
Complex numbers can be represented graphically.
Polar coordinates are useful in trigonometry and physics.

Mathematical Notation

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

$z = re^{i\theta}$ Euler's formula

$\angle z$ argument of a complex number

Remember to use proper notation when solving problems

Why It Works

Polar coordinates simplify the representation of rotations and magnitudes in complex numbers by using $r$ and $\theta$.

Remember

Convert polar to rectangular form using $x = r\cos(\theta)$ and $y = r\sin(\theta)$.

Quick Reference

Polar to Cartesian:$x = r\cos(\theta), y = r\sin(\theta)$

Cartesian to Polar:$r = \sqrt{x^2 + y^2}, \theta = \mathrm{Arg}(x + yi) = \tan^{-1}(\frac{y}{x})$ (adjust quadrant)

Understanding Polar Form of a Complex Number

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(\frac{y}{x})$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the polar form of the complex number $3 + 4i$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're plotting a point in a video game scenario using polar coordinates. The point is $7\angle 45^\circ$. What are the Cartesian coordinates (x, y) of this point?

Thinking Challenge

Thinking Exercise