16. Parametric Equations & Polar Coordinates

A polar conic section Consider the equation r² = sec2θ

b. Find the vertices, foci, directrices, and eccentricity of the curve."

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

(1, 2π/3)

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

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

Convert the point to rectangular coordinates.

$(4,\(\frac{\pi}{6}\))$

Graph $r=3\(\cos\)4\(\theta\)$

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

Convert each equation to its rectangular form.

$r=\(\frac{2}{1-\sin\theta}\)$

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

r = sin θ sec² θ

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

$r=4\(\sin\)2\(\theta\)$

Plot the point on the polar coordinate system.

$(6,-\(\frac{11\pi}{6}\))$

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.

r = 1 and r = 2 sin 2θ

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

Cartesian to Polar Equations

Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.

x - y = 3

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.