From the story. To know how high their rocket went, the boys needed a way to measure height without climbing the sky. Hickam's account says they built their own theodolites, angle-measuring sights, and used trigonometry to estimate altitude. (More in October Sky.)
Here is the idea. Measure a baseline $b$ along the ground from the launch pad to where you stand. When the rocket reaches its highest point, sight it and read the angle of elevation $\theta$ above level. If the rocket went straight up, the pad, you and the rocket make a right triangle. The baseline is the side next to the angle; the height is the side opposite it.
🔑 Key idea
$$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{b} \qquad\Longrightarrow\qquad h = b\tan\theta.$$ Add the height of the instrument above the ground. To find the angle to expect: $\theta = \tan^{-1}(h/b)$.
The tracker follows the rocket up. Try a short baseline and a long one with the same height: the angle changes, but $b\tan\theta$ comes out the same.
NASA's classroom version of this uses a hand-held tracker at a fixed baseline (30 m in their activity), adds the height of the person holding it, and averages readings from more than one station. How long a baseline to choose is the subject of the next lesson.
The tangent grows slowly at first and then without limit near $90^\circ$: at steep angles a small misreading makes a big difference in height.
✏️ Worked example: A theodolite reading
A tracker $200\ \text{m}$ from the pad, with the sight $1.5\ \text{m}$ above the ground, reads $58^\circ$ at apogee. How high was the rocket?
Add $1.5\ \text{m}$: about $322\ \text{m}$, or roughly $1{,}050$ feet.
Answer: $\approx 322\ \text{m}$
✏️ Worked example: Where to look
A rocket should reach about $400\ \text{m}$. A tracker stands $250\ \text{m}$ from the pad. At what angle should they expect it?
$\tan\theta = 400/250 = 1.6$.
$\theta = \tan^{-1}1.6 \approx 58^\circ$.
Answer: $\approx 58^\circ$
⚠️ Common mistake
Using sine instead of tangent: $\sin\theta$ is opposite over the hypotenuse, the slanted sight line, which nobody measured. Also check the calculator is in degrees: in radians mode, $\tan 58$ comes out near $8.4$, which makes no sense for this triangle.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 3 XP.