Function defined from an arc of a circle or from the value of the angle at the centre corresponding to an arc of a circle.
A function f is a circular function on a unit circle C if and only if f: \(\mathbb{R}\) → C : | t |→ (a, b) where | t | is the measure of an arc on C and (a, b) is the ordered pair of coordinates of the end point P of the arc of measure | t |.
- The “circular function” is often called a “winding function.”
- The numbers a and b are called the cosine and sine of the angle θ formed by the x-axis and the radius OP.
- The functions that align the values a and b to an angle θ of a unit circle are also circular functions.
A circular function P is a function in which every number t of the number line \(\mathbb{R} \) is made to correspond to a point P (t) on a unit circle centred on the origin of the Cartesian plane. On the unit circle C, to each arc measure, there corresponds a point of the circle:
- à | t | = 0 corresponds the ordered pair (1, 0)
- à | t | = π/2 corresponds the ordered pair (0, 1)
- à | t | = π corresponds the ordered pair (–1, 0)
- à | t | = 3π/2 corresponds the ordered pair (0, –1)
- à | t | = 2π corresponds the ordered pair (1, 0)