Unit 3: Trigonometric & Polar Functions

The Polar Coordinate System

Plotting and interpreting points and graphs in polar coordinates.

Advanced25 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Trigonometric Equations & Inequalities

Locating a Point by Angle and Distance

Every point so far has been located by two perpendicular distances, x and y. Before reading on: if you were only allowed to describe a point by how far it is from a fixed center and which direction to look, could you still pin it down exactly?

Definition — Polar Coordinates

A point in polar coordinates is written (r, θ): r is the distance from the pole (origin), and θ is the angle measured counterclockwise from the positive x-axis. Polar and rectangular coordinates describe the same plane and convert between each other with x = r·cos(θ), y = r·sin(θ), and r = √(x² + y²).

x = r\cos\theta, \qquad y = r\sin\theta

Worked Example — Converting Polar to Rectangular

Convert (r, θ) = (4, 2π/3) to rectangular form. x = 4cos(2π/3) = 4(−1/2) = −2. y = 4sin(2π/3) = 4(√3/2) = 2√3. So the rectangular point is (−2, 2√3). Check: √((−2)² + (2√3)²) = √(4 + 12) = √16 = 4, matching the original r.

Worked Example — Converting Rectangular to Polar

Convert (−3, 3) to polar form. r = √((−3)² + 3²) = √18 = 3√2. The reference angle is arctan(3/3) = arctan(1) = π/4, but (−3, 3) sits in Quadrant II (x negative, y positive), so θ = π − π/4 = 3π/4. Check: 3√2·cos(3π/4) = 3√2(−√2/2) = −3, and 3√2·sin(3π/4) = 3√2(√2/2) = 3 — both match.

A polar function r(θ) also has a rate of change, the same idea from Unit 1 applied to a new pair of variables: how r changes as θ increases tells you whether the curve is spiraling toward the pole or away from it, over that interval of angles.

θ	0	π/6	π/3	π/2	2π/3	5π/6	π
r = 2 + 2cos(θ)	4	2 + √3 ≈ 3.73	3	2	1	2 − √3 ≈ 0.27	0

r = 2 + 2cos(θ), a cardioid, sampled from θ = 0 to θ = π.

Worked Example — Average Rate of Change of a Polar Function

For r(θ) = 2 + 2cos(θ), the table above shows r falling steadily from 4 at θ = 0 to 0 at θ = π — the curve moves continuously closer to the pole as θ sweeps through the first half-turn, reaching the pole itself at θ = π. The average rate of change of r with respect to θ over [0, π] is (r(π) − r(0))/(π − 0) = (0 − 4)/π = −4/π ≈ −1.27, negative because r is decreasing overall — the same average-rate-of-change formula from Unit 1, now applied to r as a function of θ instead of y as a function of x.

Tip

A single point can have infinitely many polar representations — adding 2π to θ or using (−r, θ + π) both land on the same point — so when converting from rectangular form, state one valid (r, θ) pair rather than searching for a single 'correct' answer.

Common Mistakes

Computing θ as a bare arctan(y/x) without checking which quadrant the point is actually in.
arctan only returns angles in (−π/2, π/2) — for a point in Quadrant II or III, add π to the reference angle (as shown above), and for Quadrant IV, use a negative angle or add 2π, matching the point's actual location.
Assuming r must always be positive.
r can be negative, plotted by going the given distance in the opposite direction of θ — a fact used to reach a point 'the other way' without changing the angle, though this lesson's examples all use positive r.

Key Takeaways

A polar point (r, θ) converts to rectangular via x = r·cos(θ), y = r·sin(θ), and back via r = √(x² + y²) with θ found from the correct quadrant.
A polar function r(θ) has its own average rate of change, computed the same way as any other average rate of change, just with θ as the input variable.
A decreasing r(θ) means the curve is moving toward the pole as θ increases; an increasing r(θ) means it's moving away.

Summary

This closes Unit 3: sinusoidal modeling, trigonometric equations, and polar coordinates all describe periodic and rotational behavior. The final unit generalizes further — functions that use a third variable, a parameter, to control x and y independently, along with the vectors and matrices built on that same idea.

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