cosech function


Martin McBride, 2021-02-04
Tags cosech sinh
Categories hyperbolic functions pure mathematics

The cosech function is a hyperbolic function. It is also known as the hyperbolic cosecant function.

Equation and graph

The cosech function is the reciprocal of the sinh function, and is only defined for non-zero values:

$$ \operatorname{cosech}{x} = \frac{1}{sinh{x}} $$

(This is analogous with trig functions, where cosec is the reciprocal of sin.)

Substituting the formula for sinh gives:

$$ \operatorname{cosech}{x} = \frac{2}{e^{x}-e^{-x}} $$

Here is a graph of the function:

Sketching cosech

We can sketch the cosech curve based on the sinh curve:

The sinh curve tends to -∞ for large negative $x$, it passes through the origin at $x = 0$, and it tends to +∞ for large positive $x$.

The cosech curve is the reciprocal of sinh. It tends to zero for large negative $x$, and tends to -∞ as $x$ approaches zero from the negative direction. It also tends to zero for large positive $x$, and but tends to +∞ as $x$ approaches zero from the positive direction.

Other forms of the equation

If we take the two alternative forms of the sinh function, and invert them, we get two alternative forms of the cosech function. First:

$$ \operatorname{cosech}{x} = \frac{2e^{x}}{e^{2x}-1} $$

Alternatively, if we multiply the top and bottom of the original equation for the sinh function by $e^{-x}$ we get:

$$ \operatorname{cosech}{x} = \frac{2e^{-x}}{1-e^{-2x}} $$

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