sine

(left) The sine of an angle; (right) constructing a sine wave. The sine of an angle is a function used in the mathematical study of the triangle. If the sine of angle β is known, then the hypotenuse can be found given the length of the opposite side, or the opposite side can be found from the hypotenuse. Within a circle of unit radius (left), the height P₁A₁ equals the sine of angle P₁OA₁. This fact and the equalities below the circle allow a sine curve to be drawn, as on the right.

The sine is a function of an angle in a right-angled triangle found by dividing the length of the side opposite the angle by the length of the hypotenuse (the longest side). Sine (usually abbreviated sin) is one of the fundamental trigonometric ratios.

In trigonometry, a function of an angle in a right-angled triangle that is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse (the longest side). This function can be used to find either angles or sides in a right-angled triangle.

For example, a photographer setting up a camera on a 1 m-high tripod to take photographs of birds nesting 12 m high up in a tree, 18 m from the observation site, can work out the angle at which to set the camera:

sin H = opposite/hypotenuse = 11/18 = 0.61

so

H = 38° (to the nearest degree)

The photographer must set the camera at 38° to the horizontal.

Sines and sine rules are also used to plot sine wave and trigonometric graphs (see trigonometry).