Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r² = 4r sin θ

Lengths of Curves

Find the lengths of the curves in Exercises 25–30.

x = cos t, y = t + sin t, 0 ≤ t ≤ π

Examples of Polar Equations

[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.

r = −2 cos θ

Cartesian to Polar Equations

Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.

x - y = 3

Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the ray (half line) with initial point (-1,2) that passes through the point (0,0)

Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the line segment with endpoints (-1,3) and (3,-2)

Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π