16. Parametric Equations & Polar Coordinates

102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ>0. Let a=1 and a=−1.

Spiral of Archimedes: r = aθ

42–43. Intersection points Find the intersection points of the following curves.

r= √(cos3t) and r= √(sin3t)

Polar conversion Consider the equation r=4/(sinθ+cosθ). 

a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).

b. Explain why tan θ = y/x.

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

Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?

b. r=(½)+sinθ

A polar conic section Consider the equation r² = sec2θ

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

80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.

r² - 8r cos(θ - π/2) = 9

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.  

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

a. The point with Cartesian coordinates (−2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4), (2√2, −5π/4), and (−2√2,−π/4).