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One of the trigonometric functions:

The domain of definition is the whole real line and the range of values is the interval . The sine is an odd periodic function of period . Sine and cosine are connected by the formula

Sine and cosecant are connected by the formula

The derivative of sine is:

The indefinite integral of sine is:

Sine has the following power series representation:

The function inverse to sine is called arcsine.

The sine and cosine of a complex argument are related to the exponential function by Euler's formulas:

and if is pure imaginary, then

where is the hyperbolic sine.


Of course, can be defined by the Euler formulas or by its power series. A visual definition runs as follows. Consider the unit circle with centre at the origin in a rectangular coordinate system and with a rotating radius vector . Let be the angle between and (being reckoned positive in the counter-clockwise direction) and let be the projection of on . Then is defined as the ratio , as the ratio and as the ratio .

Figure: s085480a

Another, analytical, approach starts with the function defined on the closed interval by . For this integral is improper, but convergent. It is easy to see that is monotone increasing and continuous on the closed interval and differentiable on the open interval , and has values in . So it has an inverse function, defined on , with values in . This function is called , and it can be proved that the domain of definition of this function can be continued to the whole real axis. The function is called arcsine.

The graph of is the sinusoid (see also Trigonometric functions).


[a1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1972) pp. §4.3
How to Cite This Entry:
Sine. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Yu.A. Gor'kov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article