ASSIGNMENT 11

POLAR EQUATIONS

BY : SHARREN M. THOMAS

**THE
PROBLEM:**

Investigate
r = a + b cos (kθ)

Below
see the graphs, when a = b and k = 1 these heart-shaped graphs are called
cardioids.

r = 2
+ 2 cos θ; r = 4 + 4 cos θ; r = 5 - 5 cos θ; r = 3 - 3 cos
θ

If
a and b are not zero, then the graph r = a + b cos (kθ) where k = 1 are
limacons. When a/b < 1 the graph will be a limacon with an inner
loop. See below.

r = 2
+ 4 cos θ; r = 2 - 3 cos θ

When
1 < a/b < 2, the graph will be a limacon with a dimple. When a/b
≥ 2 then the graph will be a convex limacon. See below.

r = 6
+ 4 cos θ; r = 8 + 4 cos θ.

r
= a cos(kθ) and r = a sin(kθ) are the general for of polar equations
whose graphs are four-leafed roses. For any positive integer n greater
that 1 and any nonzero real number a has a graph that consists of a number of
loops through the origin. If n is even, there are 2n loops, and if n is
odd, there are n loops. See samples below and also press click to
manipulate with graphing calculator.

**r=a+b
cos (kΘ) where a=b and k is a noninteger**

One
notices from the graphs of r=4+4 cos (1/2(Θ)), r=4+4 cos (4/3(Θ) and r=4+4 cos (11/4(Θ))
noninteger
values of k, partial leaves occur though.

**r=a+b
cos (kΘ) where a<b and k is an odd integer smaller leaves appear inside
the larger ones.**