# TI-BASIC:Tanh

**Command Summary**

Calculates the hyperbolic tangent of a value.

**Command Syntax**

tanh(*value*)

**Menu Location**

The tanh( command is only found in the Catalog. Press:

- 2nd CATALOG to access the command catalog.
- T to skip to commands starting with T.
- Scroll down and select tanh(.

TI-83/84/+/SE

1 byte

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:

<math> \operatorname{sgn} x=\begin{cases}-1&\text{if }x<0,\\0&\text{if }x=0,\\1&\text{if }x>0.\end{cases} </math>

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: <math> \tanh{x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^{2x}-1}{e^{2x}+1} </math>