Sine (sin)

What is sine

The sine of an angle in a right triangle is the ratio of the opposite leg to the hypotenuse. On the unit circle, sine is the vertical (y) coordinate of the point.

Domain and properties

The domain is all real numbers, the range is the interval [−1, 1]. The function is periodic with period 360° (2π) and odd: sin(−α) = −sin(α).

Notable values

sin 0° = 0, sin 30° = 0.5, sin 45° = √2/2 ≈ 0.707, sin 60° = √3/2 ≈ 0.866, sin 90° = 1

Where sine is used in practice

• Physics of waves and oscillation – sound, light and alternating current are all described by a sine wave.
• Height of a point on a rotating wheel, or the height of the Sun above the horizon during the day.
• Signal processing and music – every tone can be decomposed into sine components.

The unit circle and graphs

The best way to understand trigonometric functions is the unit circle – a circle of radius 1 centred at the origin. Moving along it by an angle α, the coordinates of the point are exactly (cos α, sin α). This gives the fundamental identity sin²α + cos²α = 1 (the Pythagorean theorem for that point). The graph of the function is obtained by plotting the function value against the angle – producing the characteristic wave.

Relationships between the functions

tan α = sin α / cos α  •  cot α = 1 / tan α  •  sin²α + cos²α = 1