To solve the equation \((x - 1)^2 = \frac{9}{16}\), follow these steps:

1. Take the square root of both sides of the equation:
\[
\sqrt{(x - 1)^2} = \pm \sqrt{\frac{9}{16}}
\]

2. Simplify the square root:
\[
x - 1 = \pm \frac{3}{4}
\]

3. Solve for \(x\) by isolating it on one side of the equation:
\[
x - 1 = \frac{3}{4} \quad \text{or} \quad x - 1 = -\frac{3}{4}
\]

4. Add 1 to both sides of each equation to solve for \(x\):
\[
x = 1 + \frac{3}{4} \quad \text{or} \quad x = 1 - \frac{3}{4}
\]

5. Simplify the results:
\[
x = \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \quad \text{or} \quad x = \frac{4}{4} - \frac{3}{4} = \frac{1}{4}
\]

So, the solutions to the equation \((x - 1)^2 = \frac{9}{16}\) are:
\[
x = \frac{7}{4} \quad \text{or} \quad x = \frac{1}{4}
\]