16. Parametric Equations & Polar Coordinates

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.

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

24–26. Sets in polar coordinates Sketch the following sets of points.

4 ≤ r² ≤ 9

27–32. Polar curves Graph the following equations.

r = 3 cos 3θ

27–32. Polar curves Graph the following equations.

r = 3 sin 4θ

Channel flow Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point P(r,θ) in the channel, the flow is in the tangential direction (counterclock wise along circles), and it depends only on r, the distance from the center of the semicircles.

a. Express the region formed by the channel as a set in polar coordinates.

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

d. Plot the curve with various values of k. How many fingers can you produce?