Section 9.6

Polar Coordinates

Instead of marching East (x) and North (y), proceed a distance in direction . This makes circles and spirals simple to describe.

Understanding Polar Coordinates

The Grid vs. The Radar

In the Cartesian (Rectangular) system, you are an ant marching on a city grid. To get to point , you walk blocks East, then turn and walk blocks North.

In the Polar system, you are the operator of a radar station or a cannon. To hit a target, you don't give a grid location. Instead, you specify:

Direction (): The angle to aim, measured counter-clockwise from the positive x-axis (East).
Range (): The straight-line distance to shoot.

The Connection: It's Just Right Triangles

If we overlay the circular polar world onto the rectangular grid, every point forms a right triangle with the origin. Using SOH CAH TOA:

Polar to Rectangular

Given , find

Rectangular to Polar

Given , find

Common Pitfalls

The Quadrant Trap: could mean (Q1) or (Q3). Your calculator usually gives the Q1/Q4 answer. Always draw a picture to check which quadrant your point is in.
Negative Radius: If is negative, it means "walk backwards." The point is in the same location as —exactly opposite through the pole.

Real-World Application

Microphone Pickup Patterns: Audio engineers select microphones based on their "polar pattern." A cardioid microphone (which looks like a heart on a polar graph) picks up sound clearly from the front () and sides but blocks noise from the back (). This math ensures singers are heard while crowd noise is rejected.

Polar Rose

r = 2 cos(kθ)

k = 3

Worked Example

Converting Equations

Convert to Cartesian.

1. Multiply by r

2. Substitute

3. Complete the Square

. (Circle centered at (0,1)).

Level Up Examples

A. The Hidden Parabola

Convert to a polar equation solving for .

Step 1: Substitute and Simplify

Replace with and with :

Step 2: Solve for r

Divide by (assuming ):

Graph of

B. The Offset Circle

Convert to a Cartesian equation.

Step 1: Create Substitutions

Multiply both sides by to create terms we recognize:

Step 2: Convert and Complete Square

Substitute and :

This is a circle centered at with radius 2.