Polar Form of Complex Numbers
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Modulus of a Complex Number
Definition:
The modulus (or absolute value) of a complex number is the distance from the origin to the point in the complex plane
For \( z = x + yi \):
\[ |z| = r = \sqrt{x^2 + y^2} \]
Properties:
• The modulus is always non-negative: \( r \geq 0 \)
• \( r = 0 \) if and only if \( z = 0 \)
• Also written as \( |z| \) or \( \text{mod}(z) \)
Examples:
Find modulus of \( z = 3 + 4i \)
\( |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
Find modulus of \( z = -5 + 12i \)
\( |z| = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \)
2. Argument of a Complex Number
Definition:
The argument is the angle \( \theta \) (measured in radians) that the line from the origin to the point makes with the positive real axis
For \( z = x + yi \):
\[ \theta = \arg(z) = \tan^{-1}\left(\frac{y}{x}\right) \]
⚠️ Must consider the quadrant when finding the angle!
Finding Argument by Quadrant:
| Quadrant | Condition | Argument Formula |
|---|---|---|
| I | \( x > 0, y > 0 \) | \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \) |
| II | \( x < 0, y > 0 \) | \( \theta = \pi + \tan^{-1}\left(\frac{y}{x}\right) \) |
| III | \( x < 0, y < 0 \) | \( \theta = \pi + \tan^{-1}\left(\frac{y}{x}\right) \) |
| IV | \( x > 0, y < 0 \) | \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \) or \( 2\pi + \tan^{-1}\left(\frac{y}{x}\right) \) |
Principal Argument:
The principal argument is the unique value of \( \theta \) in the range:
\[ -\pi < \theta \leq \pi \quad \text{or} \quad -180° < \theta \leq 180° \]
3. Polar Form of Complex Numbers
Definition:
The polar form represents a complex number using its modulus and argument
\[ z = r(\cos\theta + i\sin\theta) \]
\[ \text{or} \quad z = r \text{ cis } \theta \]
\[ \text{or} \quad z = r\angle\theta \]
where:
• \( r = |z| = \sqrt{x^2 + y^2} \) (modulus)
• \( \theta = \arg(z) \) (argument)
4. Convert Rectangular to Polar Form
Conversion Formulas:
Given: \( z = x + yi \) (rectangular form)
Find: \( z = r(\cos\theta + i\sin\theta) \) (polar form)
\[ r = \sqrt{x^2 + y^2} \] \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \text{ (adjust for quadrant)} \]
Steps:
1. Calculate the modulus: \( r = \sqrt{x^2 + y^2} \)
2. Find the reference angle: \( \alpha = \tan^{-1}\left(\frac{|y|}{|x|}\right) \)
3. Determine the argument \( \theta \) based on the quadrant
4. Write in polar form: \( z = r(\cos\theta + i\sin\theta) \)
Examples:
Convert \( z = 1 + i \) to polar form
Modulus: \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \)
Argument: \( \theta = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = \frac{\pi}{4} \)
Point (1, 1) is in Quadrant I
Polar form: \( z = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) \)
Convert \( z = -\sqrt{3} + i \) to polar form
Modulus: \( r = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2 \)
Reference angle: \( \alpha = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \)
Point \( (-\sqrt{3}, 1) \) is in Quadrant II
Argument: \( \theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \)
Polar form: \( z = 2\left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right) \)
Convert \( z = 3 - 3i \) to polar form
Modulus: \( r = \sqrt{3^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2} \)
Reference angle: \( \alpha = \tan^{-1}\left(\frac{3}{3}\right) = \frac{\pi}{4} \)
Point (3, -3) is in Quadrant IV
Argument: \( \theta = -\frac{\pi}{4} \) or \( \frac{7\pi}{4} \)
Polar form: \( z = 3\sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right) \)
5. Convert Polar to Rectangular Form
Conversion Formulas:
Given: \( z = r(\cos\theta + i\sin\theta) \) (polar form)
Find: \( z = x + yi \) (rectangular form)
\[ x = r\cos\theta \] \[ y = r\sin\theta \]
Steps:
1. Evaluate \( \cos\theta \) and \( \sin\theta \)
2. Multiply by \( r \): \( x = r\cos\theta \) and \( y = r\sin\theta \)
3. Write in rectangular form: \( z = x + yi \)
Examples:
Convert \( z = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) \) to rectangular form
Evaluate: \( \cos\frac{\pi}{3} = \frac{1}{2} \) and \( \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \)
Calculate: \( x = 2 \cdot \frac{1}{2} = 1 \)
Calculate: \( y = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \)
Rectangular form: \( z = 1 + \sqrt{3}i \)
Convert \( z = 4\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right) \) to rectangular form
Evaluate: \( \cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \) and \( \sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2} \)
Calculate: \( x = 4 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -2\sqrt{2} \)
Calculate: \( y = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \)
Rectangular form: \( z = -2\sqrt{2} + 2\sqrt{2}i \)
Convert \( z = 6\left(\cos\pi + i\sin\pi\right) \) to rectangular form
Evaluate: \( \cos\pi = -1 \) and \( \sin\pi = 0 \)
Calculate: \( x = 6(-1) = -6 \)
Calculate: \( y = 6(0) = 0 \)
Rectangular form: \( z = -6 \) (pure real number)
6. Quick Conversion Reference
| From | To | Formulas |
|---|---|---|
| Rectangular \( z = x + yi \) | Polar \( z = r(\cos\theta + i\sin\theta) \) | \( r = \sqrt{x^2 + y^2} \) \( \theta = \tan^{-1}(y/x) \) |
| Polar \( z = r(\cos\theta + i\sin\theta) \) | Rectangular \( z = x + yi \) | \( x = r\cos\theta \) \( y = r\sin\theta \) |
7. Special Cases
Common Complex Numbers in Polar Form:
| Rectangular | Polar Form |
|---|---|
| \( 1 \) | \( 1(\cos 0 + i\sin 0) \) |
| \( i \) | \( 1(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}) \) |
| \( -1 \) | \( 1(\cos\pi + i\sin\pi) \) |
| \( -i \) | \( 1(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}) \) |
8. Quick Reference Summary
Essential Formulas:
Modulus: \( r = |z| = \sqrt{x^2 + y^2} \)
Argument: \( \theta = \arg(z) = \tan^{-1}(y/x) \) (adjust for quadrant)
Polar Form: \( z = r(\cos\theta + i\sin\theta) = r \text{ cis } \theta \)
Rectangular to Polar: \( r = \sqrt{x^2+y^2}, \theta = \tan^{-1}(y/x) \)
Polar to Rectangular: \( x = r\cos\theta, y = r\sin\theta \)
📚 Study Tips
✓ Always check the quadrant when finding the argument
✓ Modulus is always non-negative (like distance)
✓ Principal argument range: -π < θ ≤ π
✓ Polar form makes multiplication and division easier
✓ Draw a diagram to visualize the complex number's position