Sine

Sine, one of the fundamental ratios of trigonometry. A ratio is a proportional relationship between two numbers calculated by dividing one number by the other. Sine embodies the relationship between the lengths of the sides of a right triangle (a triangle with a 90° angle) and the magnitudes of its angles. This relationship means that varying one value, such as the length of a side, requires another value, such as the magnitude of an angle, to change in a predictable way.

The sine, usually abbreviated sin, of one of the acute (less than 90°) angles of a right triangle is equal to the length of the side opposite the acute angle divided by the length of the longest side, called the hypotenuse: .

Sine is also defined for angles greater than 90° using right triangles inscribed in a circle centered at the point (0,0) on the xy axis:

A line drawn from the circle’s center to any point on the circle makes an angle, θ, with the x axis. The sine of θ is equal to the vertical distance of the point from the x axis divided by the length of the line connecting the point to the circle’s center. Sine smoothly decreases in numerical value from 1 to -1 as θ increases from 90° to 270° and then increases again to 0 as θ goes from 270° to 360°.

Cosecant is sine’s reciprocal function. The cosecant, usually abbreviated csc, of an acute angle of a right triangle is equal to the length of the triangle’s hypotenuse divided by the length of the side opposite the chosen acute angle: . See also Cosine, Secant, Tangent, Cotangent.