Cosecant

Cosecant ($\csc$) is a trigonometric ratio. In a right triangle with an angle $\theta$ ,

$\csc(\theta)=\frac{\text{hypotenuse}}{\text{opposite}}$

$\text{Opposite}$ is the side of the triangle not part of angle $\theta$ , and $\text{hypotenuse}$ is the side opposite the right angle.

Properties

The reciprocal of cosecant is sine (abbreviated as $\sin$), while its inverse is $\operatorname {arccsc}$ or $\csc^{-1}$ . Note that cosecant is not being raised to the power of -1; this is an inverse function, not a reciprocal.

The derivative of $\csc(x)$ is $-\csc(x)\cot(x)$ , while its antiderivative is $\ln\bigl(\left|\tan(\tfrac{x}{2})\right|\bigr)$ .