AP Precalculus: Polar Form of Complex Numbers

Master modulus, argument, and conversions between rectangular and polar forms

📐 Modulus 🔄 Argument ↔️ Conversions 📊 Polar Graphs

📚 Understanding Polar Form

While rectangular form \(a + bi\) describes a complex number by its horizontal and vertical components, polar form uses distance from the origin (modulus) and angle from the positive real axis (argument). This representation is especially useful for multiplication, division, and powers of complex numbers.

1 Modulus and Argument

Every complex number \(z = a + bi\) can be described by two polar components: the modulus (distance from origin) and the argument (angle from positive real axis).

Modulus (r)

\(r = |z| = \sqrt{a^2 + b^2}\)

Argument (θ)

\(\theta = \arg(z) = \arctan\left(\frac{b}{a}\right)\)

⚠️ Quadrant Adjustment

The basic \(\arctan\left(\frac{b}{a}\right)\) only gives angles in Quadrants I and IV. You must adjust based on the signs of \(a\) and \(b\)!

Quadrant Adjustment Rules

Quadrant	Signs (a, b)	Argument Formula
I	\(a > 0, b > 0\)	\(\theta = \arctan\left(\frac{b}{a}\right)\)
II	\(a < 0, b> 0\)	\(\theta = \pi + \arctan\left(\frac{b}{a}\right)\)
III	\(a < 0, b < 0\)	\(\theta = \pi + \arctan\left(\frac{b}{a}\right)\)
IV	\(a > 0, b < 0\)	\(\theta = 2\pi + \arctan\left(\frac{b}{a}\right)\)

💡 Principal Argument

The principal argument is typically in the range \((-\pi, \pi]\) or \([0, 2\pi)\). Be consistent with which convention you use.

2 Polar Form Notation

Once you have modulus \(r\) and argument \(\theta\), you can write the complex number in polar form.

Polar Form \(z = r(\cos\theta + i\sin\theta)\)

CIS Notation (Shorthand) \(z = r \operatorname{cis} \theta\) where \(\operatorname{cis}\theta = \cos\theta + i\sin\theta\)

Euler's Form (Advanced) \(z = re^{i\theta}\)

Rectangular Form

\(z = a + bi\)
Uses x-y coordinates

Polar Form

\(z = r \operatorname{cis} \theta\)
Uses radius-angle coordinates

3 Converting Rectangular to Polar

Given \(z = a + bi\), find \(r\) and \(\theta\) to write in polar form.

Calculate modulus: \(r = \sqrt{a^2 + b^2}\)
Find reference angle: \(\alpha = \arctan\left|\frac{b}{a}\right|\)
Determine quadrant from signs of \(a\) and \(b\)
Adjust angle \(\theta\) based on quadrant
Write: \(z = r(\cos\theta + i\sin\theta)\) or \(z = r \operatorname{cis} \theta\)

📌 Example

Convert to polar: \(z = -1 + \sqrt{3}i\)

Modulus: \(r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2\)

Reference angle: \(\arctan\left|\frac{\sqrt{3}}{-1}\right| = \arctan(\sqrt{3}) = \frac{\pi}{3}\)

Quadrant: \(a < 0, b> 0\) → Quadrant II

Argument: \(\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\)

Polar form: \(z = 2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) = 2 \operatorname{cis} \frac{2\pi}{3}\)

4 Converting Polar to Rectangular

Given \(z = r \operatorname{cis} \theta\), find \(a\) and \(b\) to write in rectangular form.

Find Real Part

\(a = r\cos\theta\)

↓

Multiply modulus by cosine

Find Imaginary Part

\(b = r\sin\theta\)

↓

Multiply modulus by sine

Polar to Rectangular \(z = r(\cos\theta + i\sin\theta) = r\cos\theta + i \cdot r\sin\theta = a + bi\)

📌 Example

Convert to rectangular: \(z = 4 \operatorname{cis} \frac{\pi}{6}\)

Real part: \(a = 4\cos\frac{\pi}{6} = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}\)

Imaginary part: \(b = 4\sin\frac{\pi}{6} = 4 \cdot \frac{1}{2} = 2\)

Rectangular form: \(z = 2\sqrt{3} + 2i\)

5 Operations in Polar Form

Polar form makes multiplication, division, and powers much easier — just work with modulus and argument separately!

Multiplication

\(z_1 \cdot z_2 = r_1 r_2 \operatorname{cis}(\theta_1 + \theta_2)\)

Multiply moduli, add arguments

Division

\(\frac{z_1}{z_2} = \frac{r_1}{r_2} \operatorname{cis}(\theta_1 - \theta_2)\)

Divide moduli, subtract arguments

📌 Example: Multiplication

Given: \(z_1 = 3 \operatorname{cis} \frac{\pi}{4}\) and \(z_2 = 2 \operatorname{cis} \frac{\pi}{6}\)

Product: \(z_1 \cdot z_2 = (3)(2) \operatorname{cis}\left(\frac{\pi}{4} + \frac{\pi}{6}\right) = 6 \operatorname{cis} \frac{5\pi}{12}\)

💡 Why Polar is Powerful

In rectangular form, multiplying \((a+bi)(c+di)\) requires FOIL. In polar form, just multiply the lengths and add the angles!

6 De Moivre's Theorem (Powers)

De Moivre's Theorem provides a formula for raising a complex number to any power.

De Moivre's Theorem \([r \operatorname{cis} \theta]^n = r^n \operatorname{cis}(n\theta)\)

📌 Example

Calculate: \((1 + i)^6\)

Convert to polar: \(r = \sqrt{1^2 + 1^2} = \sqrt{2}\), \(\theta = \frac{\pi}{4}\)

So \(1 + i = \sqrt{2} \operatorname{cis} \frac{\pi}{4}\)

Apply De Moivre: \((\sqrt{2})^6 \operatorname{cis}\left(6 \cdot \frac{\pi}{4}\right) = 8 \operatorname{cis} \frac{3\pi}{2}\)

Convert back: \(8\left(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right) = 8(0 - i) = -8i\)

7 Common Polar Graphs

In polar coordinates, points are given as \((r, \theta)\). Understanding common polar equations helps visualize complex number relationships.

Circle (centered at origin)

\(r = a\) (constant)

Line through origin

\(\theta = \alpha\) (constant)

Cardioid

\(r = a(1 + \cos\theta)\)

Rose Curve

\(r = a\cos(n\theta)\)

Spiral

\(r = a\theta\)

Lemniscate

\(r^2 = a^2\cos(2\theta)\)

📋 Quick Reference

Modulus

\(r = \sqrt{a^2 + b^2}\)

Argument

\(\theta = \arctan\left(\frac{b}{a}\right)\) + quadrant adj.

Polar Form

\(r \operatorname{cis} \theta\) or \(r(\cos\theta + i\sin\theta)\)

To Rectangular

\(a = r\cos\theta\), \(b = r\sin\theta\)

Multiply/Divide

Multiply/divide \(r\), add/subtract \(\theta\)

De Moivre's

\([r \operatorname{cis} \theta]^n = r^n \operatorname{cis}(n\theta)\)

📚 Understanding Polar Form

1 Modulus and Argument

Quadrant Adjustment Rules

2 Polar Form Notation

Rectangular Form

Polar Form

3 Converting Rectangular to Polar

4 Converting Polar to Rectangular

5 Operations in Polar Form

6 De Moivre's Theorem (Powers)

7 Common Polar Graphs

📋 Quick Reference

Modulus

Argument

Polar Form

To Rectangular

Multiply/Divide

De Moivre's

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