To determine the exact value of \( \cos\left(\frac{3\pi}{4}\right) \), follow these steps:

1. Recognize that \( \frac{3\pi}{4} \) is in the second quadrant (since it's between \( \frac{\pi}{2} \) and \( \pi \)).

2. Use the reference angle for \( \frac{3\pi}{4} \), which is:
   \[
   \text{Reference angle} = \pi - \frac{3\pi}{4} = \frac{\pi}{4}
   \]
   
3. In the second quadrant, cosine values are negative, and the cosine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \).

4. Therefore:
   \[
   \cos\left(\frac{3\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}
   \]

Thus, the exact value of \( \cos\left(\frac{3\pi}{4}\right) \) is \( -\frac{\sqrt{2}}{2} \).

Would you like more details on how the reference angle or cosine values work?

Here are some related questions:
1. How do we find reference angles for different quadrants?
2. Why is the cosine negative in the second quadrant?
3. What are the sine and tangent values for \( \frac{3\pi}{4} \)?
4. How do trigonometric values change in different quadrants?
5. What is the unit circle and how does it help in finding trigonometric values?

**Tip:** When using reference angles, always remember to adjust the sign of the trigonometric function based on the quadrant.

Determine the exact value of cos(3π/4) using trigonometric principles such as reference angles and the unit circle. This problem demonstrates how cosine behaves in the second quadrant, with detailed steps for students in Grades 9-12.

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