Sine and cosine on the unit circle

Put the unit circle's centre at the origin and start at A(1, 0). Turn through an angle x (in radians) to reach P(a, b) on the circle. Then cos x = a and sin x = b: cosine is the x-coordinate of P and sine the y-coordinate.

Because P lies on the unit circle, a² + b² = 1, which gives cos² x + sin² x = 1 for every real x.

For acute x these agree with the triangle ratios of earlier classes, but the unit-circle definition works for every real x, including obtuse, reflex and negative angles.

The quadrantal angles are the integer multiples of π/2. At x = 0, π/2, π and 3π/2 the point P is (1, 0), (0, 1), (−1, 0) and (0, −1), so cos x takes the values 1, 0, −1, 0 and sin x the values 0, 1, 0, −1.

A further turn of 2π brings P back to the same point, so sin (2nπ + x) = sin x and cos (2nπ + x) = cos x for every integer n. For example, sin 750° = sin (720° + 30°) = sin 30° = 1/2.

sin x = 0 exactly when x is an integer multiple of π, that is x = nπ, n ∈ Z.

cos x = 0 exactly when x is an odd multiple of π/2, that is x = (2n + 1)π/2, n ∈ Z.