Rearranging functions/inverse relationships

[adron]

Homework Statement

Say z = f(x,y) where x = r cos(theta) and y = r sin(theta), how would I go about rearranging this to find r(x,y)?

Homework Equations

The Attempt at a Solution

I thought maybe I should find r in terms of x and y and then equate them, but I don't think that's right.. a point in the right direction would be much appreciated.

 

[Dick]

What's x^2+y^2? Find a trig identity to use on that expression in terms of r and theta.

 

[adron]

Ok, so now I have
r = (x^2 + y^2)/sqrt(cos^2 theta + sin^2 theta)

since x^2 + y^2 = 1.

Is that correct? And is that the same as r(x,y)?

 

[Dick]

Ok, so now I have
r = (x^2 + y^2)/sqrt(cos^2 theta + sin^2 theta)

since x^2 + y^2 = 1.

Is that correct? And is that the same as r(x,y)?

x^2+y^2=r^2*(sin(theta)^2+cos(theta)^2). Isn't that what you mean??

 

[adron]

x^2 + y^2 = 1 doesn't it?
so then (r cos(theta))^2 + (r sin(theta))^2 = 1 = x^2 + y^2
And then I just rearranged them to find r in terms of x,y.