Find the exact value of cosine, 150, degreescos150∘ in simplest form with a rational denominator.

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.

Learn how to find the exact value of cosine 150 degrees using trigonometric principles. Understand the concept of reference angles and cosine values in different quadrants. Suitable for high school students studying trigonometry.

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