Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: Polar coordinates (r, θ) name a point by its distance from the origin and its angle from the positive x-axis, with x = r cos θ and y = r sin θ. Parametric equations give x and y as functions of a parameter such as time, and eliminating the parameter reveals the path’s rectangular equation.
Standards: N-CN.B.4, F-IF.C.7 · Level: Grade 12 Precalculus
How do polar coordinates locate a point?
A point is named by a distance and a direction. In polar coordinates (r, θ), r is the signed distance from the pole (the origin) and θ is the angle from the polar axis (the positive x-axis). A negative r means “go the opposite way”, so (−3, 40°) and (3, 220°) are the same point, and adding 360° or 2π to θ changes nothing.
How do you convert between polar and rectangular form?
x = r cos θ, y = r sin θ, r2 = x2 + y2, and tan θ = yx for x ≠ 0.
- Multiply both sides by r if that creates an r cos θ or r sin θ term.
- Replace r cos θ with x, r sin θ with y, and r2 with x2 + y2.
- Complete the square to name the shape.
Convert r = 6 cos θ. Multiply by r: r2 = 6r cos θ, so x2 + y2 = 6x. Completing the square gives (x − 3)2 + y2 = 9, a circle with center (3, 0) and radius 3.
Which shapes do the common polar equations make?
| Equation | Graph | Key feature |
|---|---|---|
| r = a | circle | center at the pole, radius |a| |
| r = a cos θ or a sin θ | circle | passes through the pole, diameter |a| |
| r = a cos nθ or a sin nθ | rose | n petals if n is odd, 2n if even |
| r = a + b cos θ | limaçon | inner loop if a < b; cardioid if a = b |
Using tan⁻¹(yx) without checking the quadrant. For (−2, 2) the calculator gives −45°, but the point is at 135°.
What are parametric equations and how do you eliminate the parameter?
Parametric equations write both coordinates as functions of a third variable, usually time t: x = f(t) and y = g(t). To eliminate the parameter, solve one equation for t and substitute it into the other. For x = 3t and y = 12t − 4t2, t = x3, so y = 4x − 4x29, a parabola.
For a trigonometric pair such as x = 4 cos t and y = 2 sin t, do not solve for t. Use cos2 t + sin2 t = 1 instead.
- Conversion: x = r cos θ, y = r sin θ, r2 = x2 + y2.
- A point has many polar names: (−r, θ + π) and (r, θ + 2π) name the same point as (r, θ).
- Roses r = a cos nθ have n petals when n is odd and 2n when n is even.
- Limaçons r = a + b cos θ have an inner loop when a < b and a cusp when a = b.
- To eliminate a parameter, solve for t or use an identity such as cos2 t + sin2 t = 1.
Polar Coordinates and Parametric Curves: frequently asked questions
Can r be negative in polar coordinates?
Yes. A negative r means moving that distance in the direction opposite to θ. So (−5, 30°) is the same point as (5, 210°). Always check the sign when reading a point off a graph.
Why does r = 6 cos θ make a circle?
Multiplying by r gives x2 + y2 = 6x, which becomes (x − 3)2 + y2 = 9 after completing the square. The curve passes through the pole and has diameter 6.
How many petals does a rose with an even n have?
A rose r = a cos nθ or a sin nθ has 2n petals when n is even. So r = 5 sin 4θ has 8 petals, each of length 5.
How do I eliminate the parameter from x = cos t and y = sin t?
Square both equations and add: x2 + y2 = cos2 t + sin2 t = 1. The path is the unit circle, which a single equation in x and y shows.
Does the rectangular equation show the direction of motion?
No. Eliminating the parameter loses the timing and direction. Keep a table of t values or arrows on the graph when the order of travel matters, such as when a path is traced only once.
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