### Instruction: Compute $\tan 330^\circ$.### Response: I know that the tangent of an angle is the ratio of the opposite side to the adjacent side in a right triangle, or equivalently, the ratio of the sine to the cosine of the angle.
I also know that the angles in a full rotation add up to 360 degrees, so 330 degrees is the same as 360 degrees minus 30 degrees, or $360^\circ - 30^\circ$.
This means that the tangent of 330 degrees is the same as the tangent of the negative of 30 degrees, or $\tan 330^\circ = \tan (-30^\circ)$.
I can use the property that the tangent function is odd, meaning that $\tan (-\theta) = -\tan (\theta)$ for any angle $\theta$.
This implies that $\tan 330^\circ = -\tan 30^\circ$.
I remember that 30 degrees is a special angle that has an exact value for the tangent function, namely $\tan 30^\circ = \frac{1}{\sqrt{3}}$, or equivalently, $\tan 30^\circ = \frac{\sqrt{3}}{3}$ if I rationalize the denominator.
Therefore, I can multiply both sides of the equation by -1 to get the final answer: $\tan 330^\circ = -\frac{1}{\sqrt{3}}$ or $\tan 330^\circ = -\frac{\sqrt{3}}{3}$.