Evaluate. Write your answer in simplified, rationalized form. Do not round.
cos ($$ \frac{7\pi}{6})= $$

We are asked to evaluate $ \cos \left( \frac{7\pi}{6} \right) $ and express the answer in simplified, rationalized form.

### Step-by-Step Solution:

1. **Determine the Reference Angle:**
   - The angle $ \frac{7\pi}{6} $ is in radians.
   - Convert this angle into degrees: 
     \[
     \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ
     \]
   - The reference angle for $ 210^\circ $ is:
     \[
     210^\circ - 180^\circ = 30^\circ
     \]
   So the reference angle is $ 30^\circ $.

2. **Determine the Quadrant:**
   - $ \frac{7\pi}{6} $ or $ 210^\circ $ is located in the third quadrant.
   - In the third quadrant, the cosine function is **negative**.

3. **Use the Cosine Value of the Reference Angle:**
   - The cosine of $ 30^\circ $ (or $ \frac{\pi}{6} $) is:
     \[
     \cos 30^\circ = \frac{\sqrt{3}}{2}
     \]
   - Since the angle is in the third quadrant, where cosine is negative, we have:
     \[
     \cos \left( \frac{7\pi}{6} \right) = -\frac{\sqrt{3}}{2}
     \]

### Final Answer:
\[
\cos \left( \frac{7\pi}{6} \right) = -\frac{\sqrt{3}}{2}
\]

Would you like further details or have any questions?

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Here are 5 related questions:
1. What is the sine of $ \frac{7\pi}{6} $?
2. How do you convert an angle from radians to degrees?
3. What is the cosine of $ \frac{5\pi}{6} $?
4. How do the signs of trigonometric functions change in different quadrants?
5. What is the value of $ \sin \left( \frac{\pi}{6} \right) $?

**Tip:** When dealing with trigonometric functions, always determine the reference angle and consider the sign based on the quadrant!

Evaluate. Write your answer in simplified, rationalized form. Do not round. cos(7π/6) = 

Learn how to evaluate cos(7π/6) using trigonometric concepts, reference angles, and quadrant rules. This step-by-step solution simplifies the cosine function and is perfect for students in Grades 10-12 learning about radians and the unit circle.

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