16. Parametric Equations & Polar Coordinates

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

r = 6 cos θ + 8 sin θ

Plot the point on the polar coordinate system.

$(-3,-90°)$

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.

(4, 5π)

Polar conversion Consider the equation r=4/(sinθ+cosθ). 

a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.

r = 2 - 2 sin θ  b

Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?

b. r=(½)+sinθ

Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.

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

42–43. Intersection points Find the intersection points of the following curves.

r= √(cos3t) and r= √(sin3t)

(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933

a. Show that f(0) = f(2π) and find the point on the curve that corresponds to θ = 0 and θ = 2π.