Cosecant

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:

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

The series expansion is:

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

Comments

See also Sine.