Use reference angles to find the exact value of the following expression. \[ \cos \frac{3 \pi}{4} \]

1. Express the angle in standard position. The angle given is \( \frac{3\pi}{4} \). 2. Determine the quadrant where \( \frac{3\pi}{4} \) lies. Since \( \frac{3\pi}{4} \) is between \( \frac{\pi}{2} \) and \( \pi \), it is in the second quadrant. 3. Find the reference angle. The reference angle for an angle in the second quadrant is found by subtracting the angle from \( \pi \): \[ \text{Reference angle} = \pi - \frac{3\pi}{4} = \frac{\pi}{4}. \] 4. Evaluate the cosine of the reference angle. We know that: \[ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}. \] 5. Adjust the sign based on the quadrant. In the second quadrant, cosine is negative, so: \[ \cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2}. \] Thus, the exact value is \( -\frac{\sqrt{2}}{2} \).