AP Precalculus: Polar Form of Complex Numbers & Conversions

1. Modulus and Argument

Given \( z = a + bi \):
Modulus (absolute value): \( r = |z| = \sqrt{a^2 + b^2} \)
Argument (angle): \( \theta = \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \)
(Adjust for quadrant; use \( \arctan2(b,a) \) if available)

2. Rectangular to Polar Form

Rectangular: \( z = a + bi \)
Polar: \( z = r\left(\cos\theta + i\sin\theta\right) \) or \( z = r\operatorname{cis} \theta \)
Find \( r \) and \( \theta \) as above; plug in to write in polar form.

3. Polar to Rectangular Form

If \( z = r(\cos\theta + i\sin\theta) \), then
- Real part: \( a = r\cos\theta \)
- Imaginary part: \( b = r\sin\theta \)
- Rectangular: \( z = a + bi \)

4. Converting Between Forms

Rectangular to Polar: Find \( r, \theta \) as above and write \( z = r\operatorname{cis}\theta \)
Polar to Rectangular: \( z = r\cos\theta + i r\sin\theta \)
"cis" notation: \( \operatorname{cis}\theta = \cos\theta + i\sin\theta \)

5. Match Polar Equations & Graphs (Basics)

Points in polar: \( (r, \theta) \)
Complex numbers in polar: on plane, angle \( \theta \) from positive real axis, modulus as radius
Common polar graphs: circles (\( r = a \)), lines (\( \theta = \alpha \)), spirals, roses etc.