Explicit proof of the Jacobian inverse

1. Apr 30, 2016

Physgeek64

1. The problem statement, all variables and given/known data
Given the transformations $x^2+y^2=2*r*cos(theta)$ and $x*y=r*sin(theta)$ prove the Jacobian explicitly

The question then goes on to ask how r and theta are related to the cylindrical coordinates rho and phi. I think $r=1/2*(x^2+y^2)$ and hence $r=1/2 rho$ but Im not really sure about this part I'm afraid, so haven't gotten very far

2. Relevant equations
$(partial(x,y)/partial(r,theta))*partial(r,theta)/partial(x,y))=1$

3. The attempt at a solution
So by multiplying the second expression by two and then squaring both and adding we get $r=1/2*(x^2+y^2)$ from which we can find the first two elements in the second Jacobian to be x and y respectively. By dividing the two transformations we can also get an expression for theta, which is relativity simple to differentiate, however when isolating x and y I seem to be going into pages of algebra, which leads me to think I have got the wrong approach, and there must be a simpler way since this is meant to be a relatively quick question.

Many thanks

2. May 5, 2016