The unit circle

What is the unit circle?

The unit circle is a circle with centre O and radius 1, situated in an orthonormal co-ordinate system (O; I, J). It is assigned a positive direction of travel, known as the direct direction (anti-clockwise). The other direction is the indirect (or negative) direction.

The four reference points

We often denote them as follows:

Point	Coordinates	Associated real number (first revolution)
I	(1, 0)	0
J	(0, 1)	π/2
I'	(-1, 0)	π
J'	(0, -1)	3π/2

These four points divide the circle into four quarters (quadrants), which are useful for quickly determining the sign of the cosine and sine of a point.

What is this circle used for?

Every point M on the circle has coordinates (x, y) such that x² + y² = 1 (the equation of a circle of radius 1). These coordinates precisely define the cosine and sine: x = cos(t) and y = sin(t), where t is a real number associated with point M.

Common pitfall

Do not confuse the radius of the unit circle (which is always equal to 1) with that of any other circle: it is precisely because the radius is 1 that cos(t) and sin(t) correspond directly to the coordinates of the point, without any multiplicative factor.