How do you find the exact value of tan 150 degrees?

To find the exact value of (\tan 150^\circ), you can use the fact that (150^\circ) is in the second quadrant, where the tangent function is negative. The reference angle for (150^\circ) is (180^\circ - 150^\circ = 30^\circ). Therefore, (\tan 150^\circ = -\tan 30^\circ). Since (\tan 30^\circ = \frac{1}{\sqrt{3}}), it follows that (\tan 150^\circ = -\frac{1}{\sqrt{3}}), or (-\frac{\sqrt{3}}{3}) when rationalized.