16. Parametric Equations & Polar Coordinates

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?

What is the polar equation of a circle of radius √(a²+b²) centered at (a, b)?

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

Navigating A plane is 150 miles north of a radar station, and 30 minutes later it is 60 degree east of north at a distance of 100 miles from the radar station. Assume the plane flies on a straight line and maintains constant altitude during this 30-minute period.

a. Find the distance traveled during this 30-minute period.

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

Explain why the slope of the line θ=π/2 is undefined.

Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.

Cartesian conversion Write the equation x=y ² in polar coordinates and state values of θ that produce the entire graph of the parabola.

(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.