To evaluate $\cos 150^\circ$ without using a calculator, we can use a reference triangle. Here's how to do it step by step:

1. Identify the reference angle:
   $150^\circ$ is in the second quadrant. To find the reference angle, subtract $150^\circ$ from $180^\circ$: $180^\circ - 150^\circ = 30^\circ$ So, the reference angle is $30^\circ$.

2. Determine the cosine of the reference angle:
   In the first quadrant, $\cos 30^\circ = \frac{\sqrt{3}}{2}$. This is a well-known trigonometric ratio.

3. Consider the sign of cosine in the second quadrant:
   In the second quadrant, cosine is negative. Therefore, we apply the negative sign to the cosine of the reference angle: $\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$

Thus, the value of $\cos 150^\circ$ is $-\frac{\sqrt{3}}{2}$.

Would you like a detailed explanation of the unit circle or more examples? Here are some related questions for further practice:

1. What is the cosine of $120^\circ$?
2. How can you find the sine of $150^\circ$?
3. What are the coordinates of the point on the unit circle corresponding to $150^\circ$?
4. How does the reference angle help in evaluating trigonometric functions for angles in different quadrants?
5. How would you evaluate $\cos 210^\circ$ using a reference triangle?

Tip: When evaluating trigonometric functions of angles in the second quadrant, remember that sine is positive and cosine is negative.