### Instruction: Compute $\tan 30^\circ$.### Response: I recall that $\tan \theta$ is defined as the ratio of the opposite side to the adjacent side in a right triangle with angle $\theta$.
One way to find $\tan 30^\circ$ is to use a special right triangle that has a $30^\circ$ angle, such as an equilateral triangle cut in half.
If I draw such a triangle, I can label the sides as follows: the hypotenuse is 2, the side opposite the $30^\circ$ angle is 1, and the side adjacent to the $30^\circ$ angle is $\sqrt{3}$, by the Pythagorean theorem.
Therefore, $\tan 30^\circ$ is the ratio of 1 to $\sqrt{3}$, or $\frac{1}{\sqrt{3}}$.
I could also rationalize the denominator by multiplying the fraction by $\frac{\sqrt{3}}{\sqrt{3}}$, which gives me $\frac{\sqrt{3}}{3}$.