# Sine

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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.