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.