y = \csc x

Graph of the Cosecant Function $y = \csc x$

The cosecant function $y = \csc x$ is the trigonometric function defined as the reciprocal of the sine¹. It is also written $\operatorname{cosec} x$ , and in a right triangle it is the ratio of the hypotenuse to the opposite side.

Definition

\csc x = \frac{1}{\sin x}

Domain and range

The domain is every real number with $x \neq n\pi$
The range is $y \leq -1$ or $y \geq 1$
The period is $2\pi$
It is an odd function

Since $-1 \leq \sin x \leq 1$ and $\sin x \neq 0$ , we have $|\csc x| \geq 1$ .

Symmetry and period

Because $\sin x$ is odd, $\csc(-x) = -\csc x$ , so the graph has point symmetry about the origin. The period is $2\pi$ , the same as the sine.

Asymptotes and limits

There is a vertical asymptote at each $x = n\pi$ .

Approach	$\sin x$	$\csc x$
$x \to 0^{+}$	$0^{+}$	$+\infty$
$x \to 0^{-}$	$0^{-}$	$-\infty$

Monotonicity and extrema

The derivative is $\dfrac{d}{dx}\csc x = -\csc x \cot x$ . Between consecutive asymptotes the curve forms a U or an inverted U.

$x$	$\sin x$	$\csc x$	Extremum
$\dfrac{\pi}{2} + 2n\pi$	$1$	$1$	local minimum
$\dfrac{3\pi}{2} + 2n\pi$	$-1$	$-1$	local maximum

Relationships with other functions

1 + \cot^2 x = \csc^2 x

It is tied to the secant by $\csc x = \sec\left( \dfrac{\pi}{2} - x \right)$ ; shifting the graph of $\sec x$ right by $\dfrac{\pi}{2}$ produces $\csc x$ .

Item	$\sec x$	$\csc x$
Definition	$\dfrac{1}{\cos x}$	$\dfrac{1}{\sin x}$
Asymptotes	$\dfrac{\pi}{2} + n\pi$	$n\pi$
Symmetry	even	odd
Derivative	$\sec x\tan x$	$-\csc x\cot x$

Integral

\int \csc x\,dx = \ln\left|\tan\frac{x}{2}\right| + C

Applications

Triangle calculations that use the law of sines in reciprocal form
The analysis of oscillations and waves
Integrals in which $\csc^2 x$ appears as the derivative of the cotangent

Trigonometric functions, Wikipedia

Graph of the Cosecant Function y=csc⁡xy = \csc xy=cscx