Solving any triangle means calculating its main elements: $a$, $b$, $c$, $\hat{A}$, $\hat{B}$ and $\hat{C}$. To do this, it is necessary to provide three information about the triangle, at least one of which is the measurement of a segment linked to it (side, height, median, etc.). The story The first European treatise dedicated entirely to solving triangles, both plane and spherical, was De triangulis omnimodis (“On triangles of all types”), written around 1464 by the German astronomer Regiomontanus and published only in 1533. He organized the solving cases in a very similar way to what we will do here, and the book was the gateway to trigonometry in Renaissance Europe.
Solving a right-angled triangle means calculating its main elements: the acute angles ($\hat{B}$ and $\hat{C}$) and the sides ($a$, $b$, $c$). To do this, it is necessary to provide two information about the triangle, at least one of which is the measurement of a segment connected to it (side, height, median, angle bisector etc.) — only with angles, we know the shape but not the size. The story Solving right triangles was, for centuries, the daily work of surveyors, navigators and artillerymen. To measure angles on the ground, the Englishman Leonard Digges described an instrument he called theodelitus — the ancestor of the theodolite, which to this day appears on the streets on a tripod. The description appeared in Pantometria (“Universal Measurement”), a book that Leonard left unfinished and that his son, Thomas Digges, completed and published in 1571. With a measured angle and a known distance, the trigonometric tables did the rest.