The tanh() Command

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Command Summary

Takes the hyperbolic tangent of a number.

Command Syntax

tanh(value)

Menu Location

• Press 2nd MATH to enter the MATH menu.
• Press C to enter the Hyperbolic submenu.
• Press 3 to select tanh(.

Calculator Compatibility

This command works on all calculators.

Token Size

1 byte

The tanh() command returns the hyperbolic tangent of a number.

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

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

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

If tanh() is applied to a list, it will take the hyperbolic tangent of every element in the list.

Advanced Uses

The tanh() of a matrix is not (in general) the same as taking the hyperbolic tangent of every element of the matrix. A different definition is used to compute the result; see Matrices and Their Commands. It requires the matrix to be square and diagonalizable in order to apply.

Formulas

The definition of hyperbolic cotangent is, by analogy with tan(), the ratio of sinh() and cosh():

(1)

\begin{align} \tanh{x}=\frac{\sinh{x}}{\cosh{x}} = \frac{e^x-e^{-x}}{e^x+e^{-x}} = \frac{e^{2x}-1}{e^{2x}+1} \end{align}

Error Conditions

230 - Dimension happens when taking tanh() of a matrix that isn't square.

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

665 - Matrix not diagonalizable happens when taking tanh() of a matrix that isn't diagonalizable.

Related Commands

• sinh()
• cosh()
• tanhֿ¹()