In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously.

If we imagine a point moved back and forth synchronously with a pendulum, and if such point made a mark upon paper, it would trace the same line over and over again. If now the paper were drawn steadily along at right angles to the line of motion of the point, then the point would trace upon it a line like the profile of a wave. Such line is a sine curve. It derives its name from the following construction. Let a straight line be drawn, and laid off in fractions, such as degrees, of the perimeter of a circle of given diameter. Then on each division of the line let a perpendicular be erected equal in height to the sine of the angle of the circle corresponding to that division; then if the extremities of such lines be united by a curve such curve will be a sine curve. In such a curve the abscissas are proportional to the times, while the ordinates are proportional to the sines of angles, which angles are themselves proportional to the times. The ordinates pass through positive and negative values alternately, while the abscissas are always positive. Any number of sine curves can be constructed by varying the diameter of the original circle, or by giving to the abscissas a value which is a multiple of the true length of the divisions of circle. If the pendulum method of construction were used this would be attained by giving a greater or less velocity to the paper as drawn under the pendulum. A species of equation for the curve is given as follows: y = sin( x ) In this x really indicates the arc whose length is x, and reference should be made to the value of the radius of the circle from which the curve is described. It will also be noticed that the equation only covers the case in which the true divisions of the circle are laid off on the line. If a multiple of such divisions are used, say n times, or 1-n times, then the equation should read y = n sin( x ) or y = sin( x ) / n Synonyms--Curve of Sines--Sinusoidal Curve--Harmonic Curve.