1. **State the problem:** Find the angle $x$ such that $\cos x = \frac{1}{4}$.\n\n2. **Formula and rules:** The cosine function relates an angle to the ratio of the adjacent side over the hypotenuse in a right triangle. To find $x$, use the inverse cosine function: $$x = \cos^{-1}\left(\frac{1}{4}\right).$$\n\n3. **Calculate the angle:** Using a calculator or inverse cosine table, evaluate $$x = \cos^{-1}\left(\frac{1}{4}\right).$$\n\n4. **Result:** The principal value (in radians) is approximately $$x \approx 1.3181.$$\n\n5. **General solution:** Since cosine is positive in the first and fourth quadrants, the general solutions are $$x = \pm \cos^{-1}\left(\frac{1}{4}\right) + 2k\pi, \quad k \in \mathbb{Z}.$$\n\nThus, the angle $x$ satisfying $\cos x = \frac{1}{4}$ is approximately $1.3181$ radians plus full rotations.