9.7 Defining Polar Coordinates and Differentiating in Polar Form

Polar coordinates basics

A point in the polar coordinate system is written as (r, θ), where r is the distance from the origin (pole) and θ is the angle from the positive x-axis. Conversions: x = r cos θ, y = r sin θ, and r² = x² + y², tan θ = y/x. A polar curve is often given as r = f(θ).

Finding dy/dx for polar curves

Since x and y are both functions of θ, use parametric form: x = f(θ)cos θ, y = f(θ)sin θ. Then dy/dx = (dy/dθ)/(dx/dθ), computed via product rule on each. This slope tells you the tangent line's steepness at a given θ, useful for finding horizontal/vertical tangents (set numerator or denominator to 0).

Horizontal & vertical tangents

A horizontal tangent occurs where dy/dθ = 0 (and dx/dθ ≠ 0); a vertical tangent occurs where dx/dθ = 0 (and dy/dθ ≠ 0). Always check both derivatives aren't simultaneously zero, and verify using the original r = f(θ) equation to avoid extraneous points from the conversion.

Key terms

Polar coordinate (r, θ)

r = distance from origin, θ = angle from positive x-axis.

Polar-to-rectangular conversion

x = r cos θ, y = r sin θ; r² = x² + y², tan θ = y/x.

dy/dx for polar curve r = f(θ)

(dy/dθ)/(dx/dθ), where x = f(θ)cos θ and y = f(θ)sin θ (use product rule for each derivative).

Horizontal tangent condition

dy/dθ = 0 and dx/dθ ≠ 0.

Vertical tangent condition

dx/dθ = 0 and dy/dθ ≠ 0.

Try an AP-style question

Convert the polar point (r, θ) = (2, π/3) to rectangular coordinates.

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