Property	Value
Hex Value	`$CC`
Categories	Catalog > T
Localizations	FR: `th(`

`tanh(`

Overview

Returns hyperbolic tangent of a real number, expression, or list.

Availability: Token available everywhere.

Syntax

tanh(value)

Arguments

Name	Type	Optional
value	real\|expression\|real[]

Location

2nd, catalog, tanh(

Description

Calculates the hyperbolic tangent of a value. The hyperbolic trig functions sinh(, cosh(, and tanh( are an analog of normal trig functions, but for a hyperbola, rather than a circle. They can be expressed in terms of real powers of e, and don't depend on the Degree or Radian mode setting.

tanh(0)
    0
tanh(1)
    .761594156

Like normal trig commands, tanh( works on lists as well, but not on complex numbers, even though the function is often extended to the complex numbers in mathematics.

Advanced Uses

The tanh( command can be used to approximate the sign function:

(1) $\begin{align} \texttt{sgn} x=\begin{cases}-1&\text{if }x<0,\\0&\text{if }x=0,\\1&\text{if }x>0.\end{cases} \end{align}$

As the absolute value of the input becomes large, the convergence is achieved at a point closer to zero. For the function to work as intended generally, numbers having lesser orders of magnitude need to be multiplied by a factor large enough for the argument to arrive at ±16.720082053122, which is the smallest input to produce ±1 (respectively) to fourteen digits of accuracy.

5/12→X
    .4166666667
tanh(E9X)
    1
tanh(-E9X)
    -1

Formulas

The definition of the hyperbolic tangent is:

(2) $\begin{align} \tanh{x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^{2x}-1}{e^{2x}+1} \end{align}$

Source: parts of this page were written by the following TI|BD contributors: burr, DarkerLine, Edward H, GoVegan, thornahawk, Timothy Foster, Weregoose.

History

Calculator	OS Version	Description
TI-82	1.0	`tanh` added
TI-83	0.01013	Renamed `tanh` to `tanh(`

Authors: Adrien Bertrand

tanh(