The coth() Command

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Command Summary

Takes the hyperbolic cotangent of a number.

Command Syntax

coth(value)

Menu Location

• Press 2nd MATH to enter the MATH menu.
• Press C to enter the Hyperbolic submenu.
• Press 6 to select coth(.

Calculator Compatibility

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)

Token Size

1 byte

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

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

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

:coth(0)
           undef
:expand(coth(x))
          -1/(e^x+1)+1/(e^x-1)+1
:comDenom(1-2/((e^x)^2+1))
          1/tanh(x)

If coth() is applied to a list, it will take the hyperbolic cotangent of every element in the list. However, it can't be applied to matrices the way tanh() 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 cotangent is, by analogy with cot(), the ratio of cosh() and sinh():

(1)

\begin{align} \coth{x}=\frac{\cosh{x}}{\sinh{x}} = \frac{e^x+e^{-x}}{e^x-e^{-x}} \end{align}

Error Conditions

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

Related Commands

• sech()
• csch()
• cothֿ¹()