From the story. The boys' theodolites were homemade, so every angle they read came with some uncertainty. A good scientist reports how far off the answer might be, not just the answer. (More in October Sky.)
Suppose the angle could be off by $\pm 2^\circ$. The true height is somewhere between $b\tan(\theta - 2^\circ)$ and $b\tan(\theta + 2^\circ)$. Because the tangent climbs steeply near $90^\circ$, the same $2^\circ$ slip matters much more at steep angles.
Key idea
Height range: $b\tan(\theta - \Delta\theta) \le h \le b\tan(\theta + \Delta\theta)$. With calculus, $dh = b\sec^2\theta\,d\theta$ ($d\theta$ in radians), so the relative error $\dfrac{dh}{h} = \dfrac{d\theta}{\sin\theta\cos\theta}$ is smallest at $\theta = 45^\circ$. Two or three trackers in different directions, averaged, cancel some of the error from a rocket that drifted.
Worked example: How sure?
A tracker $100\ \text{m}$ away reads $70^\circ \pm 2^\circ$. Give the range of heights.
- $100\tan 68^\circ \approx 248\ \text{m}$; $100\tan 72^\circ \approx 308\ \text{m}$.
- So $h$ is between about $248$ and $308\ \text{m}$, a spread of $60\ \text{m}$. At $45^\circ \pm 2^\circ$ the spread would be only about $14\ \text{m}$.
Answer: $248$ to $308\ \text{m}$
Common mistake
Assuming a fixed angle error gives a fixed height error. It depends strongly on the angle.