Sine Function

The simplest and most intuitive definition of the sine function. It basically says that, on a right triangle, the following measurements are related:

• the length of the triangle's hypotenuse
• the length of one of the other sides
• the measurement of the angle (q) opposite to that other side

sin(q) = opposite / hypotenuse

This equation says that if we evaluate the sine of that angle q, we will get the exact same value as if we divided the length of the side opposite to that angle by the length of the triangle's hypotenuse. The relation holds for any right triangle, regardless of size. In mathematics, the sine (trigonometric) functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios of coordinates of points on the unit circle, or, more generally still, as infinite series, or equally generally, as solutions of certain differential equations.

All four approaches will be presented below. In all of these cases referring to triangles, the triangles are taken to exist in the Euclidean plane, so that the angles always sum to 180°.

In modern usage, there are six basic trigonometric functions, which are tabulated below along with equations relating them to one another. (Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can equally define them geometrically or by other means and derive the relations.)

These are the six basic trigonometric functions, together with their standard notational abbreviations. The last four functions are defined in terms of the first two. In other words, the four equations below are definitions, not proved identities.

• sine (sin)
• cosine (cos)
• tangent (tan = sin / cos)
• secant (sec = 1 / cos)
• cosecant (csc = 1 / sin)
• cotangent (cot = cos / sin)

Several relations between these functions are listed on the page about trigonometric identities.

A few other functions were common historically (and appeared in the earliest tables), but are now little-used, such as the versed sine and the exsecant .