### Instruction: Compute $\tan 150^\circ$.### Response: To find the tangent of $150^\circ$, I need to know the sine and cosine of this angle.
I recall that $150^\circ$ is a special angle that is in the second quadrant and has a reference angle of $30^\circ$.
I also remember the trigonometric values of $30^\circ$ from the half-equilateral triangle: $\sin 30^\circ = \frac{1}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$.
Since $150^\circ$ is in the second quadrant, its cosine is negative and its sine is positive, so $\sin 150^\circ = \frac{1}{2}$ and $\cos 150^\circ = -\frac{\sqrt{3}}{2}$.
Now I can use the definition of tangent: $\tan \theta = \frac{\sin \theta}{\cos \theta}$.
Plugging in the values I found, I get $\tan 150^\circ = \frac{\sin 150^\circ}{\cos 150^\circ} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}$.
This is the exact value of $\tan 150^\circ$, but I can also rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$.
This gives me $\tan 150^\circ = -\frac{\sqrt{3}}{3}$, which is an equivalent form of the answer.