Trigonometry and indirect measurements

Many quantities cannot be measured directly: the height of the Eiffel Tower, the distance of a ship on the horizon, the angle of the Sun above the horizon. However, they can be calculated using trigonometry: it is enough to measure something easier — the shadow of an object, a distance on the ground, an angle with an inclinometer — and apply the relations for right-angled triangles.

Key formula — Right-angled triangle

Given a right-angled triangle with hypotenuse $i$, opposite side $o$ and adjacent side $a$ with respect to angle $\alpha$: $\sin\alpha = \frac{o}{i} \qquad \cos\alpha = \frac{a}{i} \qquad \tan\alpha = \frac{o}{a}$

The strategy is always the same: identify a right-angled triangle in which an unknown quantity is a side (or the hypotenuse), measure an accessible side and an angle, and obtain the unknown by inverting the appropriate trigonometric relation. The tangent is particularly useful when both sides are known, because it links the angle directly to their ratio without needing the hypotenuse.

Historical context — Eratosthenes and the circumference of the Earth

In 240 BC the Greek Eratosthenes, librarian of Alexandria in Egypt, knew that at noon on the summer solstice the Sun lit the bottom of a well vertically at Syene (today Aswan), south of Alexandria. At the same instant, in Alexandria, an obelisk cast a shadow corresponding to an angle of $7{,}2^\circ$ from the vertical.

If the distance Syene–Alexandria was about 5000 stadia, and the angle difference amounted to $7{,}2^\circ/360^\circ = 1/50$ of a full turn, then the circumference of the Earth had to be $50 \cdot 5000 = 250\,000$ stadia, i.e. about $40\,000\;\text{km}$. The correct value is $40\,075\;\text{km}$: a surprisingly precise estimate, obtained with the instruments of antiquity alone (Simonyi 2012, ch. 1; see Riferimenti bibliografici).

Collegamenti

Argomenti: Grandezze fisiche e misura Competenze: Uso della trigonometria

Esercizi collegati: Esercizio svolto — Il rubinetto che perde · Esercizio svolto — Area di un rettangolo misurato · Problema — Volume di un parallelepipedo