16. Parametric Equations & Polar Coordinates

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

Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.

What is the polar equation of the horizontal line y = 5?

Given three polar coordinate representations for the origin.

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.

(-1, -π/3)

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.

(-4, 3π/2)

15–22. Sets in polar coordinates Sketch the following sets of points.

r = 3

15–22. Sets in polar coordinates Sketch the following sets of points.

2 ≤ r ≤ 8

23–24. Radar Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane, typically measured in degrees clockwise from north.

A plane is 50 miles from a radar station at an angle of 10 dgeree clockwise from north. Find polar coordinates for the location of the plane.

53–56. Simple curves Tabulate and plot enough points to sketch a graph of the following equations.

r = 1 - cos θ

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.

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

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

r² = 4 sin θ  

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

r = sin 3θ

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