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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 $ \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) $ .

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