Capítulo I · Neste capítulo você vai

Arcs and angles

  • recognize circle arcs, including the null arc and the arc of a turn;
  • measure arcs in degrees and radians and convert one unit to the other;
  • understand why the circumference measures $2\pi$ rad;
  • associate each central angle with the measure of its arc and calculate $\alpha = \dfrac{\ell}{r}$;
  • mark in the trigonometric cycle the image of any real number;
  • recognize congruous numbers modulo $2\pi$.

Trigonometry begins with a simple question: how to measure a turn, or a piece of a turn? The hands of a clock, a bicycle wheel and the Sun in the sky travel around circles. To work with them we need two tools: a unit to measure arcs and angles and a cycle in which each real number corresponds to a point. This chapter builds both.

Hipparchus of Nicaea (c. 190–120 BC), who divided the circumference into 360 parts in his chord tables. Later artistic representation.
Hipparchus of Nicaea (c. 190–120 BC), who divided the circumference into 360 parts in his chord tables. Later artistic representation.William Henry Smyth · Wikimedia Commons · Public domain
Leonhard Euler (1707–1783), who made the circle of radius 1 and the measurement of arcs by length standard.
Leonhard Euler (1707–1783), who made the circle of radius 1 and the measurement of arcs by length standard.Jakob Emanuel Handmann · Wikimedia Commons · Public domain

For a first conversation about spins and degrees, with paper and string activities, see also the class The mathematics of spins.