Simulate the measure of a spherical cap

Estimate by simulation the fraction of the uniform measure on $S^{d-1}$ lying in the cap $\{u : \langle u,\theta\rangle \ge t\}$, for $d \in \{3, 10, 100, 1000\}$ and $t\in\{0.1, 0.3, 0.5\}$. Compare with the bound $e^{-dt^2/2}$ and with the exact density $\propto(1-s^2)^{(d-3)/2}$. Explain why naive Monte Carlo fails for large $d$ and moderate $t$, and what to do about it.