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Cosecant

From Encyclopedia of Mathematics
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One of the trigonometric functions:

$$y=\operatorname{cosec}x=\frac{1}{\sin x};$$

other notations are $\csc x$, $\operatorname{cosc}x$. The domain of definition is the entire real line with the exception of the points with abscissas

$$x=\pi n,\quad n=0,\pm1,\pm2,\mathinner{\ldotp\ldotp\ldotp\ldotp}$$

The cosecant is an unbounded odd periodic function (with period $2\pi$). Its derivative is:

$$(\operatorname{cosec}x)'=-\frac{\cos x}{\sin^2x}=-\operatorname{cotg}x\operatorname{cosec}x.$$

The integral of the cosecant is:

$$\int\operatorname{cosec}xdx=\ln\left|\operatorname{tg}\frac x2\right|+C.$$

The series expansion is:

$$\operatorname{cosec}x=\frac1x+\frac x6+\frac{7x^3}{360}+\frac{31x^5}{15120}+\dots,\quad0<|x|<\pi.$$


Comments

See also Sine.

How to Cite This Entry:
Cosecant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cosecant&oldid=33254
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