16. Parametric Equations & Polar Coordinates

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

$r^2=-25\(\cos\)2\(\theta\)$

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.

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 = 2 cos θ and r = 1 + cos θ

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

(-4, 4√3)

Cartesian to Polar Equations

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

(x + 2)² + (y − 5)² = 16"

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

d. The point (3,π/2) lies on the graph of r=3 cos 2θ.  

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

4 ≤ r² ≤ 9

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

Graph $r=2-2\(\cos\]\theta\)$

Lines

Sketch the lines in Exercises 45–48 and find Cartesian equations for them.

r cos (θ + π/3) = 2

Convert the point to polar coordinates.

$(-1,-\(\sqrt{3}\))$

Convert each equation to its rectangular form.

$r=-4\(\cos\]\theta\)$

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.

y = 3

Convert each equation to its polar form.

$x^2+y^2=2y$

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