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Polar Equation

Polar Equation

An equation for a curve written in terms of the polar coordinates r and θ.

 

Graph of r = cos2θ, a four-petaled rose curve with petals extending along horizontal and vertical axes.

Key Formula

r=f(θ)r = f(\theta)
Where:
  • rr = The distance from the origin (pole) to a point on the curve
  • θ\theta = The angle measured counterclockwise from the positive x-axis (polar axis)
  • f(θ)f(\theta) = A function that gives r for each value of θ

Worked Example

Problem: Convert the polar equation r = 4cos(θ) to Cartesian form and identify the curve.
Step 1: Multiply both sides by r to create expressions we can convert.
r2=4rcos⁡θr^2 = 4r\cos\theta
Step 2: Use the conversion identities: r² = x² + y² and r cos θ = x.
x2+y2=4xx^2 + y^2 = 4x
Step 3: Rearrange and complete the square for x.
x2−4x+y2=0  ⟹  (x−2)2+y2=4x^2 - 4x + y^2 = 0 \implies (x - 2)^2 + y^2 = 4
Step 4: Recognize the Cartesian equation as a circle with center (2, 0) and radius 2.
(x−2)2+y2=22(x-2)^2 + y^2 = 2^2
Answer: The polar equation r = 4cos(θ) represents a circle centered at (2, 0) with radius 2.

Another Example

Problem: Plot several points for the polar equation r = 2 + 2cos(θ) and identify the type of curve.
Step 1: Evaluate r at key angles. At θ = 0: r = 2 + 2(1) = 4.
r(0)=2+2cos⁡(0)=4r(0) = 2 + 2\cos(0) = 4
Step 2: At θ = π/2: r = 2 + 2(0) = 2. At θ = π: r = 2 + 2(−1) = 0.
r ⁣(π2)=2,r(π)=0r\!\left(\tfrac{\pi}{2}\right) = 2,\quad r(\pi) = 0
Step 3: At θ = 3π/2: r = 2 + 2(0) = 2. The curve touches the origin at θ = π and extends farthest (r = 4) at θ = 0.
r ⁣(3π2)=2r\!\left(\tfrac{3\pi}{2}\right) = 2
Step 4: This equation has the form r = a + a cos θ, which produces a cardioid — a heart-shaped curve.
r=a(1+cos⁡θ),a=2r = a(1 + \cos\theta), \quad a = 2
Answer: The polar equation r = 2 + 2cos(θ) is a cardioid that passes through the origin and reaches a maximum distance of 4 from the pole.

Frequently Asked Questions

How do you convert a polar equation to a Cartesian (rectangular) equation?
Use the relationships x = r cos θ, y = r sin θ, and r² = x² + y². Substitute these into the polar equation and simplify. A common trick is to multiply both sides by r first so that r cos θ or r sin θ appears, which you can replace directly with x or y.
What are the most common types of polar equations?
Common polar curves include circles (r = a or r = a cos θ), cardioids (r = a + a cos θ), limaçons (r = a + b cos θ), rose curves (r = a cos(nθ)), and spirals (r = aθ). Each form produces a distinctive shape that is often easier to describe in polar form than in Cartesian form.

Polar equation vs. Cartesian (rectangular) equation

A polar equation expresses a curve using distance r and angle θ from a central point. A Cartesian equation uses horizontal and vertical distances x and y from perpendicular axes. Some curves, like spirals and roses, have simple polar equations but complicated Cartesian forms. Conversely, straight lines (other than those through the origin) are usually simpler in Cartesian form. The two forms describe the same curves — they just use different coordinate systems.

Why It Matters

Polar equations let you describe curves that have rotational symmetry — spirals, flowers, circles — far more naturally than Cartesian equations can. They appear throughout physics and engineering whenever problems involve rotation, orbits, or waves radiating from a point. Learning to work with polar equations also builds the foundation for parametric equations, complex numbers in polar form, and multivariable calculus.

Common Mistakes

Mistake: Forgetting that r can be negative in a polar equation and discarding those values.
Correction: A negative r means the point is plotted in the opposite direction from angle θ. For example, r = −2 at θ = 0 places the point at (2, π). Keep negative r values — they are part of the curve.
Mistake: Converting to Cartesian form by replacing r with √(x² + y²) everywhere without simplifying first.
Correction: Before substituting, multiply or rearrange so that r², r cos θ, or r sin θ appear directly. This avoids messy square roots. For instance, from r = 3 sin θ, multiply both sides by r to get r² = 3r sin θ, then substitute x² + y² = 3y.

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