Chapter 13, Trigonometric Ratios and Functions - 13.2 Define General Angles and Use Radian Measure - 13.2 Exercises - Problem Solving - Page 864: 53b

$360^{\circ}$

Given that there are $16$ steps and each step in the spiral staircase is with the central angle $\frac{\pi}{8}$. So the total angle covered by the staircase $=16\times \frac{\pi}{8}=2\pi$ that is, $360^{\circ}$. Therefore a person climbing the spiral staircase will rotate at an angle of $360^{\circ}$.