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

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

Definition

Domain and range

Where $\sin x = 0$, at $x = n\pi$ (with $n$ an integer), the denominator vanishes and the function is undefined. Its domain is all real numbers except these points. Since $-1 \le \sin x \le 1$ and $\sin x \ne 0$, we have $|\csc x| \ge 1$, so the range is $y \le -1$ or $y \ge 1$.

Symmetry and period

Because $\sin x$ is odd, $\csc(-x) = -\csc x$, so $y = \csc x$ is an odd function with point symmetry about the origin. Its period is $2\pi$, the same as sine.

Asymptotes and limits

There is a vertical asymptote at each $x = n\pi$. As $x$ approaches $0$ from the right, $\sin x \to 0^{+}$ so $\csc x \to +\infty$; from the left, $\csc x \to -\infty$.

Behavior and notable points

Between consecutive asymptotes the curve forms a $U$ shape or an inverted $U$. Where $\sin x$ attains $1$, at $x = \dfrac{\pi}{2} + 2n\pi$, $\csc x$ has a local minimum of $1$; where $\sin x$ attains $-1$, at $x = \dfrac{3\pi}{2} + 2n\pi$, $\csc x$ has a local maximum of $-1$. Its derivative is $\dfrac{d}{dx}\csc x = -\csc x \cot x$.

Relation to other functions

It relates to the secant by $\csc x = \sec\left(\dfrac{\pi}{2} - x\right)$ and appears in the identity $1 + \cot^2 x = \csc^2 x$. Shifting the graph of $\sec x$ left by $\dfrac{\pi}{2}$ gives $\csc x$.

Applications

The cosecant arises in the law of sines, in the analysis of oscillations and waves, and in integrals such as $\int \csc x\,dx = \ln\left|\tan\dfrac{x}{2}\right| + C$.