1. **State the problem:** Find the value of $\sec \frac{3\pi}{4}$. 2. **Recall the definition:** The secant function is the reciprocal of the cosine function: $$\sec x = \frac{1}{\cos x}$$ 3. **Evaluate the cosine:** Calculate $\cos \frac{3\pi}{4}$. Since $\frac{3\pi}{4} = 135^\circ$, which lies in the second quadrant where cosine is negative, and the reference angle is $\frac{\pi}{4}$: $$\cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$ 4. **Calculate secant:** $$\sec \frac{3\pi}{4} = \frac{1}{\cos \frac{3\pi}{4}} = \frac{1}{-\frac{\sqrt{2}}{2}}$$ 5. **Simplify the fraction:** $$\sec \frac{3\pi}{4} = \frac{1}{-\frac{\sqrt{2}}{2}} = \frac{1}{1} \times \frac{2}{-\sqrt{2}} = \frac{2}{-\sqrt{2}}$$ 6. **Rationalize the denominator:** $$\sec \frac{3\pi}{4} = \frac{2}{-\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{-2} = -\sqrt{2}$$ **Final answer:** $$\boxed{-\sqrt{2}}$$