it was a typo for y expression, it should be,
y=r\sin(\theta)
For the final P(r), I'm trying to find the other way that involve both \sigma_r and \sigma_\theta. So it seems like it is impossible unless i substitute them into the following expression?
P(r) = \int\int...
After doing various searching through the google, most of the circular error probability I found are expressed interm of \sigma_x and \sigma_y, which the CEP(circular error probability) usually looks like (assume normal distribution):
P(r) = \int\int...