With the reductions from the previous topic, it is enough to know the functions in the 1st quadrant. A final symmetry — in angle bisector of the 1st quadrant — reduces the work even further: it is enough to know the arcs from $0$ to $\frac{\pi}{4}$, that is, from $0^\circ$ to $45^\circ$. The story The first trigonometric tables were strings: Hipparchus and Ptolemy tabulated the string of each bow, half a degree by half. The Indians began to tabulate the half chord (the sine), and the Arab astronomers added tangents and cotangents, linked to the length of the shadows on sundials. In the 15th century, Regiomontanus calculated sines with 7 digits, and in 1596 the posthumous work of Rheticus, completed by Valentin Otho, brought the six functions of 10 in 10 seconds of arc, with up to 10 numerals — a decades-long work that calculators made unnecessary, but which underpinned navigation and astronomy for centuries. The symmetry of this topic was the first trick of those who assembled a board: calculating the sine up to $45^\circ$, the rest came for free.