The csch() Command

Command Summary

Takes the hyperbolic cosecant of a number.

Command Syntax

csch(value)

• Press 2nd MATH to enter the MATH menu.
• Press C to enter the Hyperbolic submenu.
• Press 4 to select csch(.

This command requires a calculator with AMS version 2.07 or higher (it will also work on any TI-89 Titanium or Voyage 200 calculator)

1 byte

The csch() command returns the hyperbolic cosecant of a number. Along with 11 other trig and hyperbolic functions, it was added in AMS version 2.07; on earlier versions, csch(x) can be replaced by 1/sinh(x).

As long as the calculator is in radian mode, csch() can be used with complex numbers according to the rule that csch(ix)=-csc(x)*i and csc(ix)=-csch(x)*i. This rule only works in radian mode, and csch() of a complex number will return a domain error when working in degrees or gradians.

Occasionally, csch() can compute an exact result; most of the time, the calculator will leave an expression with csch() alone unless it's in approximate mode (or you force an approximation). When csch() is used with symbolic expressions, the calculator can go back and forth between the csch() expression and its exponential equivalent.

:csch(0)
undef
:expand(csch(x))
1/(e^x+1)+1/(e^x-1)
:comDenom(1/(e^x+1)+1/(e^x-1))
1/sinh(x)


If csch() is applied to a list, it will take the hyperbolic cosecant of every element in the list. However, it can't be applied to matrices the way sinh() can (this is probably an oversight; all the trig and hyperbolic functions that were present in all AMS versions work with matrices, but the ones added in version 2.07 do not).

# Formulas

The definition of hyperbolic cosecant is, by analogy with csc(), the reciprocal of sinh():

(1)
\begin{align} \operatorname{csch}{x}=\frac{1}{\sinh{x}} = \frac{2}{e^x-e^{-x}} \end{align}

# Error Conditions

260 - Domain error happens when taking csch() of a complex number in degree or gradian mode.