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.

r² = 4r sin θ

Convert each equation to its polar form.

$3y-5x=2$

Plot the point $(5,-\(\frac{\pi}{3}\))$, then identify which of the following sets of coordinates is the same point.

102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ>0. Let a=1 and a=−1.

Spiral of Archimedes: r = aθ

80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.

r² - 8r cos(θ - π/2) = 9

Cartesian lemniscate Find the equation in Cartesian coordinates of the lemniscate r² = a² cos 2θ, where a is a real number.

What is the polar equation of the vertical line x = 5?

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.

r = 3 csc θ

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

Identify whether the given equation is that of a cardioid, limaçon, rose, or lemniscate.

$r=3+2\(\cos\]\theta\)$

Convert the point to polar coordinates.

$(0,5)$

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.

y = x²

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

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

Polar to Cartesian Coordinates

Find the Cartesian coordinates of the following points, given in polar coordinates.

c. (0, π/2)