Namespaces
Variants
Actions

Sine

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

One of the trigonometric functions:

$$y=\sin x.$$

The domain of definition is the whole real line and the range of values is the interval $[-1,1]$. The sine is an odd periodic function of period $2\pi$. Sine and cosine are connected by the formula

$$\sin^2x+\cos^2x=1.$$

Sine and cosecant are connected by the formula

$$\sin x=\frac{1}{\operatorname{cosec}x}.$$

The derivative of sine is:

$$(\sin x)'=\cos x.$$

The indefinite integral of sine is:

$$\int\sin(x)\,dx=-\cos x+C.$$

Sine has the following power series representation:

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dotsb,\qquad-\infty<x<\infty.$$

The function inverse to sine is called arcsine.

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

$$e^{iz}=\cos z+i\sin z,\\\sin z=\frac{e^{iz}-e^{-iz}}{2i},$$

and if $z=ix$ is pure imaginary, then

$$\sin ix=-\sinh x,$$

where $\sinh x$ is the hyperbolic sine.


Comments

Of course, $\sin x$ 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 $O$ in a rectangular coordinate system and with a rotating radius vector $OP$. Let $x$ be the angle between $OA$ and $OP$ (being reckoned positive in the counter-clockwise direction) and let $P'$ be the projection of $P$ on $OA$. Then $\sin x$ is defined as the ratio $(PP')/(OP)$, $\cos x$ as the ratio $(OP')/(OP)$ and $\tan x$ as the ratio $(PP')/(OP')$.

Figure: s085480a

Another, analytical, approach starts with the function $\phi$ defined on the closed interval $[-1,1]$ by $\phi(x)=\int_0^xdt/\sqrt{1-t^2}$. For $x=\pm1$ this integral is improper, but convergent. It is easy to see that $\phi$ is monotone increasing and continuous on the closed interval $[-1,1]$ and differentiable on the open interval $(-1,1)$, and has values in $[-\pi/2,\pi/2]$. So it has an inverse function, defined on $[-\pi/2,\pi/2]$, with values in $[-1,1]$. This function is called $\sin x$, and it can be proved that the domain of definition of this function can be continued to the whole real axis. The function $\phi$ is called arcsine.

The graph of $\sin x$ is the sinusoid (see also Trigonometric functions).

References

[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: http://encyclopediaofmath.org/index.php?title=Sine&oldid=44643
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