16. Parametric Equations & Polar Coordinates

Polar to Cartesian Coordinates

Find the Cartesian coordinates of the following points, given in polar coordinates.

c. (0, π/2)

What is the slope of the line θ=π/3?

Graphing Sets of Polar Coordinate Points

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises 11–26.

θ = π/2, r ≥ 0

Convert the point to rectangular coordinates.

$(-2,-\(\frac{\pi}{4}\))$

Polar to Cartesian Equations

Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.

r cos (θ − 3π/4) = (√2)/2

Polar valentine Liz wants to show her love for Jake by passing him a valentine on her graphing calculator. Sketch each of the following curves and determine which one Liz should use to get a heart-shaped curve.

c. r = cos 3θ

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.

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

(-1, -π/3)

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

(x - 1)² + y² = 1

27–32. Polar curves Graph the following equations.

r = 3 sin 4θ

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.

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

1 < r < 2 and π/6 ≤ θ ≤ π/3

Cartesian to Polar Coordinates

Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.

b. (-3,0)

Circles

Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.

r = −2 cos θ

(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?