To find the exact value of $\cos 150^\circ$, we first need to determine the reference angle and the quadrant in which $150^\circ$ lies.

Step 1: Determine the reference angle and quadrant

1. Identify the quadrant:

   • The angle $150^\circ$ is in the second quadrant since it is between $90^\circ$ and $180^\circ$.

2. Find the reference angle:

   • The reference angle for an angle in the second quadrant is given by:

   $180^\circ - 150^\circ = 30^\circ$

Step 2: Evaluate $\cos 150^\circ$

In the second quadrant, the cosine function is negative. The cosine of the reference angle $30^\circ$ is:

$\cos 30^\circ = \frac{\sqrt{3}}{2}$

Since we are in the second quadrant where cosine is negative:

$\cos 150^\circ = -\frac{\sqrt{3}}{2}$

Conclusion

The exact value of $\cos 150^\circ$ in its simplest form is:

$\cos 150^\circ = -\frac{\sqrt{3}}{2}$

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What is the value of $\sin 150^\circ$?
2. How do you find the cosine of an angle in the third quadrant?
3. What are the exact values of sine and cosine for angles like $45^\circ$ and $60^\circ$?
4. How do trigonometric functions behave in different quadrants?
5. What is the significance of reference angles in trigonometry?

Tip: When solving for trigonometric functions, always consider both the reference angle and the quadrant to determine the correct sign.