16. Parametric Equations & Polar Coordinates

Polar to Cartesian Equations

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

r = 3 cos θ

Cartesian to Polar Equations

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

x - y = 3

Cartesian to Polar Equations

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

(x + 2)² + (y − 5)² = 16"

Polar Coordinates

Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.

e = 1/3, r sin θ = −6

Graphing Sets of Polar Coordinate Points

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises 11–26.

θ = π/2, r ≥ 0

Symmetries and Polar Graphs

Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.

r = 1 + 2 sin θ

Lines

Sketch the lines in Exercises 45–48 and find Cartesian equations for them.

r cos (θ + π/3) = 2

Circles

Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.

r = −2 cos θ

Polar to Cartesian Equations

Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.

r cos (θ − 3π/4) = (√2)/2

Cartesian to Polar Equations

Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.

x² + y² + 5y = 0