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.

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:

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.

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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