Identify each polar curve and state whether it is a ........

Question

Identify each polar curve and state whether it is a rose, a line, a circle, a lima¸con with a loop, a

cardioid, a lima¸con with a dent, a convex lima¸con, a lemniscate, or none of these.

(a) r = 4

(b) r = 2 − sin θ

(c) r = 3(1 + sin θ)

(d) r = 1 − 2 sin θ

(e) r = 4 cos θ

(f) r = cos 4θ

(g) θ = 2π/3

(h) r = 4 + 3 cos θ

(i) r^ 2 = cos 2θ

(j) r = 3 − 3 cos θ

Answer 1

(a) r = 4 is a circle with a radius of 4, centered at the origin

(b) r = 2 − sin θ is a convex limacon since a/b=2/1≥2

(c) r = 3(1 + sin θ) is a cardioid since it's equivalent to r = 3 + 3sin θ and a/b=3/3=1

(d) r = 1 − 2 sin θ is an inner loop limacon since a/b=1/2<1

(e) r = 4 cos θ is a circle with radius 4 centered at (2,0)

(f) r = cos 4θ is a rose with 8 petals since 4 is even and gets doubled

(g) θ = 2π/3 is none of these

(h) r = 4 + 3 cos θ is a dimpled limacon

(i) r² = cos 2θ is a lemniscate

(j) r = 3 − 3 cos θ is a cardioid

Hope this helped!