### Instruction: Compute $\tan 210^\circ$.### 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 the slope of the line that makes that angle with the positive x-axis.
I also know that the angles on the unit circle are measured in radians, and that $180^\circ$ is equivalent to $\pi$ radians.
So, to convert $210^\circ$ to radians, I multiply by $\frac{\pi}{180}$ and get $\frac{7\pi}{6}$.
Now, I can use the unit circle to find the coordinates of the point where the line that makes an angle of $\frac{7\pi}{6}$ radians with the positive x-axis intersects the circle.
That point is $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$.
The slope of that line is the tangent of the angle, so I can find it by dividing the y-coordinate by the x-coordinate.
That gives me $\tan 210^\circ = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}$.
I could also rationalize the denominator by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$ and get $\frac{\sqrt{3}}{3}$, but either form is acceptable.