An Exact Angle

The special values run backwards too. If a ratio comes out as the entry in its own function's column of the table, the acute angle is exact: no calculator, no rounding.

A right triangle has legs of 3 and $3\sqrt{3}$. For the angle $\theta$ opposite the longer leg, $\tan\left(\theta\right) = \frac{3\sqrt{3}}{3}$, which is √3. The table says tan 60° = √3, so $\theta$ is exactly 60°. A ratio like 5/8, which is not in the table, gives an angle that has to be rounded.