From the story. A rocket rarely went perfectly straight up, which spoils the single-tracker triangle. Hickam's account describes the boys tracking flights with trigonometry, and with two sightings you no longer need to know exactly where the rocket was over the ground. (More in October Sky.)
Two observers stand a known distance $d$ apart, in line with the rocket's path. Each reads an angle of elevation at the same moment. Write each observer's ground distance to the point under the rocket as $h\cot$ of their angle. Those two distances differ by $d$ (same side) or add to $d$ (opposite sides).
Key idea
Same side (far angle $\alpha$, near angle $\beta$): $$h = \frac{d}{\cot\alpha - \cot\beta}.$$ Opposite sides: $h = \dfrac{d}{\cot\alpha + \cot\beta}$. Check with the law of sines: in the triangle of the two observers and the rocket, the angle at the rocket is $\beta - \alpha$ (same side).
Worked example: Two trackers
Two trackers on the same side, $150\ \text{m}$ apart, read $35^\circ$ (far) and $55^\circ$ (near). Find the height.
- $\cot 35^\circ = 1.428$, $\cot 55^\circ = 0.700$.
- $h = \dfrac{150}{1.428 - 0.700} \approx 206\ \text{m}$.
- Law of sines check: the near tracker's sight line is $\dfrac{150\sin 35^\circ}{\sin 20^\circ} \approx 251.5\ \text{m}$, and $251.5\sin 55^\circ \approx 206\ \text{m}$.
Answer: $\approx 206\ \text{m}$
Common mistake
Swapping the near and far angles (the denominator turns negative), or adding cotangents when both stand on the same side.