Curvature of polar function r=4cos(3θ): Find Solution

[meson0731]

Homework Statement

Given the polar function r = 4cos(3θ) find the curvature.

Homework Equations

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.

 

[tiny-tim]

hi meson0731!

use standard trigonometric identities to get:

x = 2cos(4θ) + 2cos(2θ)

y = … ? ​

 

[Filip Larsen]

There sure are plenty of trigonometric relations to choose from [1], so which have you tried? It looks like the prosthaphaeresis or product-to-sum relations may be useful. Also, if you later want to check your work using the polar formulation of curvature, it is mentioned on Weissteins page for the topic [2]. [1] http://en.wikipedia.org/wiki/List_of_trigonometric_identities
[2] http://mathworld.wolfram.com/Curvature.html