Takes the hyperbolic cotangent of a number.

coth(*value*)

**Menu Location**

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

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)

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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(** i**x)=-cot(x)

*****and cot(

*i***x)=-coth(x)**

*i******. This rule only works in radian mode, and

*i*`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()`:

# Error Conditions

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