3.11 Trigonometry and Polar Coordinates

Polar Coordinate Basics

A polar coordinate point is written as (r, θ), where r is the directed distance from the origin (pole) and θ is the angle from the positive polar axis. Unlike rectangular coordinates, a single point has infinitely many polar representations, since adding multiples of 2π to θ or negating r (and adding π to θ) gives the same point.

Converting Between Coordinate Systems

Use x = r cos(θ) and y = r sin(θ) to convert polar to rectangular. To go rectangular to polar, use r² = x² + y² and tan(θ) = y/x, being careful to choose θ in the correct quadrant. These formulas link the two systems through basic right triangle trigonometry.

Negative r and Multiple Representations

When r is negative, plot the point in the direction opposite θ. The point (r, θ) is the same as (-r, θ + π) and (r, θ + 2πk) for any integer k. This means recognizing equivalent points requires checking both r's sign and coterminal angles.

Key terms

Polar coordinates (r, θ)

r = directed distance from the pole; θ = angle from the positive polar axis.

Polar to rectangular conversion

x = r cos(θ), y = r sin(θ)

Rectangular to polar conversion

r² = x² + y², tan(θ) = y/x (choose θ based on the correct quadrant)

Negative r rule

Plot in the direction opposite θ; (r, θ) = (-r, θ + π)

Non-uniqueness of polar coordinates

Every point has infinitely many representations: (r, θ + 2πk) or (-r, θ + π + 2πk) for any integer k.

Try an AP-style question

Convert the polar point (4, π/6) to rectangular coordinates.

1. (4, π/6)
2. (2√3, 2)
3. (√3, 1)
4. (2, 2√3)