3.7: Polar Functions | AP Precalculus | Code Scholars Tutoring

How it works

In a polar function, radius changes with angle. Increasing radius means motion outward only when the radius is nonnegative; a negative radius places a point in the opposite direction. Evaluate signed radii and convert selected points before describing the curve.

r=f(\theta),\quad(x,y)=(f(\theta)\cos\theta,f(\theta)\sin\theta)

Worked examples

Example 1

For $\displaystyle r=2\cos\theta$ , find the radius at $\displaystyle \theta=\pi/3$ and convert the point.

The radius is $\displaystyle 1$ , giving Cartesian point $\displaystyle (1/2,\sqrt{3}/2)$ .

Example 2

For $\displaystyle r=1+\cos\theta$ , find the radii at $\displaystyle 0$ and $\displaystyle \pi$ .

They are $\displaystyle 2$ and $\displaystyle 0$ , so the curve reaches the pole at $\displaystyle \pi$ .

Problems

Problem 3.7.1

For $\displaystyle r=2\cos\theta$ , find the radius and Cartesian point at $\displaystyle \theta=2\pi/3$ .

A circle centered at 1,0 with radius 1, passing through the pole and 2,0. — Signed radii trace a circle even when the radius is negative.

Show worked solution

The radius is $\displaystyle -1$ . The Cartesian point is $\displaystyle (1/2,-\sqrt{3}/2)$ , opposite the ray of angle $\displaystyle 2\pi/3$ .

A circle centered at 1,0 with radius 1, passing through the pole and 2,0. The solution adds labeled relationships. — The Cartesian equation is (x - 1)^2 + y^2 = 1.

Problem 3.7.2

For $\displaystyle r=3\sin\theta$ , find the point at $\displaystyle \theta=\pi/2$ .

Show worked solution

The radius is $\displaystyle 3$ , so the Cartesian point is $\displaystyle (0,3)$ .

Problem 3.7.3

Convert $\displaystyle r=2\cos\theta$ to a Cartesian circle equation.

Show worked solution

Multiply by r: $\displaystyle x^2+y^2=2x$ , hence $\displaystyle (x-1)^2+y^2=1$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 3.7.1

Problem 3.7.2

Problem 3.7.3

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