# Inverse of a Multiple Variable Function

• Apr 21st 2008, 06:51 PM
merzbow
Inverse of a Multiple Variable Function
Ok here's the problem:
F: R^2 -> R^2
(t is theta)
F(r,t) = (r*cos(t), r*sin(t))

What is F^-1(0,0)?
I know the answer is {(0,0) | -inf < t < inf }

but I'm not exactly sure how to get to that point.
• Apr 21st 2008, 07:12 PM
TheEmptySet
Quote:

Originally Posted by merzbow
Ok here's the problem:
F: R^2 -> R^2
(t is theta)
F(r,t) = (r*cos(t), r*sin(t))

What is F^-1(0,0)?
I know the answer is {(0,0) | -inf < t < inf }

but I'm not exactly sure how to get to that point.

$F(r,\theta)=(r\cos(\theta),r\sin(\theta))$

consider $F(r,\theta)=(0,0)$

so we would need to know when

$r\cos(\theta)=0 \mbox{ and } r\sin(\theta)=0$

since cosine and sine are never zero at the same value of theta r must be zero. If that is the case then the equation

$0\cos(\theta)=0 \mbox{ and } 0\sin(\theta)=0$

is true for all values of theta so the inverse immage of (0,0) is

$[{(r,\theta)|r=0, -\infty < \theta < \infty}]$
• Apr 21st 2008, 07:18 PM
merzbow
Ah, thanks man. I knew it was something easy, I just couldn't figure it out.