Let $X_1,...,X_n$ be i.i.d. random variables with $E(X_i)=\mu$ and $V(X_i)=\sigma^2$. Suppose we are to test $H_0 : \mu=0$ against $H_1 : \mu >0$, we try to base our test statistic on $\tau_n = \frac{\sqrt{n}\bar{X}}{s_n}$ where $s_n$ is the sample standard deviation. Again, see that $\tau_n$ can be written as $\sqrt{n-1}\cot \theta$ where $\theta$ is the angle between $(1,....,1)'/ \sqrt{n}$ and $(X_1,...,X_n)$.
It can also be observed that $\tau_n$ is a monotonic decreasing function of $\theta$. My question is can we find the p-value expressed in terms of the observed $\theta$, say $\theta_0$?