Coordinate system transformation

[hjel0743]

Can someone help me with the conversion of this equation to Cartesian coordinates:

2cosθr + sinθθ

(Due to formatting limitations, I just made the r_hat and theta_hat components bold-faced)

I know the answer ought to be -(3y²)/[(x²+y²)+1] but I've tried every variation of the 3 main coordinate transformation eqns that I can think of and haven't gotten anywhere. Those 3 eqns I'm talking about are y=rsinθ, x=rcosθ, and r=sqrt(x²+y²).

Any help would be great. (Not homework related, need to get this for some research I'm working on.)

 

[HallsofIvy]

The first thing I would do is convert \hat{r} and \hat{\theta} to i and j: for a point whose radial line makes angle \theta with the positive x-axis, \hat{r}= cos(\theta)\hat{i}+ sin(\theta)\hat{j} and \hat{\theta}= -sin(\theta)\hat{i}+ cos(\theta)\hat{j}.

So 2cos(\theta)\hat{r}+ sin(\theta)\hat(\theta)= 2cos(\theta)(cos(\theta)\hat{i}+ sin(\theta)\hat{j})+ sin(\theta)(sin(\theta)\hat{i}+ cos(\theta)\hat{j}= (2cos^2(\theta)- sin^2(\theta))\hat{i}+ 3cos(\theta)sin(\theta)\hat{j}

Now, convert those trig functions into x, y.