# Curvature problem

1. Jul 7, 2012

### meson0731

1. The problem statement, all variables and given/known data
Given the polar function r = 4cos(3θ) find the curvature.

2. Relevant equations

3. The attempt at a solution

I know there is a formula for curvature of a polar function but I was never given that equation and was told to convert to parametric. and use ||v x a|| / ||v||^3. So this is my work
Converting to parametric:

x = 4cos(3θ)cos(θ)
y= 4cos(3θ)sin(θ)
z = 0

to get v i took the derivative

x' = -12sin(3θ)cos(θ) - 4cos(3θ)sin(θ)
y' = -12sin(3θ)sin(θ) + 4 cos(4θ)cos(θ)
z' = 0

I then took the second derivative to get a

x'' = -40cos(3θ)cos(θ) + 24sin(3θ)sin(θ)
y'' = -40cos(3θ)sin(θ) - 24sin(3θ)cos(θ)
z'' = 0

However I feel like i did something wrong because to computer ||v x a|| would be really messy. Is there any way to simplify before I do the cross product or is there is easier way and I'm just doing the problem wrong? Thanks any help is appreciated.

2. Jul 8, 2012

3. Jul 8, 2012