Capítulo V · Neste capítulo você vai

Notable arcs

  • demonstrate that $\operatorname{sen}\dfrac{\pi}{n}$ is half the side of the regular polygon with $n$ sides inscribed in the cycle;
  • calculate the sides of equilateral triangle, the square and the inscribed regular hexagon;
  • obtain the exact sine, cosine and tangent values ​​of $\dfrac{\pi}{6}$, $\dfrac{\pi}{4}$ and $\dfrac{\pi}{3}$;
  • extend the method to $\dfrac{\pi}{12}$ and $\dfrac{\pi}{8}$.

Almost all values ​​of trigonometric functions are irrational numbers that we only know by approximation. Some arcs, however, have exact and simple values: they are the notable arcs, linked to equilateral triangle, the square and the regular hexagon. The table that closes this chapter is the most used in all of trigonometry.

Claudius Ptolemy calculated the chords of 36°, 60°, 72°, 90° and 120° from the inscribed regular polygons. Later artistic representation.
Claudius Ptolemy calculated the chords of 36°, 60°, 72°, 90° and 120° from the inscribed regular polygons. Later artistic representation.Justus van Gent or Pedro Berruguete · Wikimedia Commons · Public domain

For a first approach with folds and triangles, see also the class Triangles and polygons.