Consider $\tan^{-1}(2+\sqrt3)$. From this one can figure out that

$$\sin(\theta)=\frac{1+\sqrt3}{2\sqrt2}$$ and

$$\cos(\theta)=\frac{\sqrt3-1}{2\sqrt2}$$

So we now know the sides of the right angle. However, if we try to use the laws of sines or cosines we will run into an obstacle, since we will, again, need to determine the values of trigonometric expressions.

So how can we determine, without using a calculator, the value of $\theta$ in this case?