Simulation · Maths · Class 11

Sine and cosine on the unit circle

From the lesson Sine and cosine on the unit circle in Trigonometric Functions. Change the values and watch what happens.

The idea behind it

NCERT §3.3

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.

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