Estimate vector $\mu=(\mu_1,\mu_2)$ if $x_i\sim \mathcal{N}(\mu_i,1)$ for $i=1,2$.

Question

Assume $x_i\sim \mathcal{N}(\mu_i,1)$ for $i=1,2$ and we are going to estimate vector $\mu=(\mu_1,\mu_2)$. If $\mu$ is located at a distance of 2 from the origin, then $\mu=(2\cos\theta,2\sin\theta)$ and the problem is to estimate $\theta$. For this let $\hat{\theta}=\arctan(x_2/x_1)$ and $-\pi<\hat{\theta}-\theta<\pi$. Now the problem is to show $\hat{\theta}$ is an unbiased estimator of $\theta$ and calculating $\mathbb{E}(\hat{\theta}-\theta)^2$ and then calculating error conditioned on $r=\sqrt{x_1^2+x_2^2}$. $\\$ I think if $P=(x_1,x_2)=(\hat{r},\hat{\theta})$ since $\mathbb{E}[x_1]=\mu_1$ and $\mathbb{E}[x_2]=\mu_2$ then intuitively $\frac{x_1}{x_2}$ is an unbiased estimation of $\frac{\mu_1}{\mu_2}$ but I couldn't prove it.