TANH()

Syntax. TANH(number)

Definition. This function returns the hyperbolic tangent of a number.

Argument

• number (required) Any real number

Background. The hyperbolic tangent belongs to the hyperbolic functions, which are (like the circle functions) defined for all real and complex numbers. (However, Excel allows only real arguments, not complex arguments, for hyperbolic functions.)

The formula for the hyperbolic tangent is:

The last term shows the similarity with the trigonometry functions.

The hyperbolic tangent is often used for research and development in engineering and natural sciences (see Figure 16-40).

Figure 16-40. The hyperbolic tangent.

You might miss the corresponding cotangent you know from the trigonometry functions: sin, cos, tan, and cot. However, neither the trigonometric cotangent nor the hyperbolic cotangent is implemented as a table function, because cotangent = tangent^–1:

• Cotangent:

  

• Hyperbolic cotangent:

  

From this, it follows that the hyperbolic cotangent is:

Instead of

=COTH(A1)

(COTH doesn’t exist), enter

=1/TANH(A1)

Example. The hyperbolic tangent is used to calculate the propagation speed of waves. The formula for the propagation speed υ of waves is:

This formula uses the gravity acceleration g [m/s²], wave length λ [m], and water depth h [m].

If you enter this formula into a table, you can use the following two approximate formulas for shallow and deep water:

• Shallow water:

  

• Deep water:

  

More examples of this function are:

• =TANH(0) returns 0.

• =TANH(1) returns 0.761594156.

• =TANH(-1) returns -0.76159416.

• =TANH(10) returns 1.

• =TANH(-10) returns -1.

See Also

ACOSH(), ASIN(), ATANH(), COSH(), SINH()

Sample Files

Use the Data&Chart and Water waves worksheets in the TanH.xls or TanH.xlsx sample file. The sample files are found in the Chapter16 folder. For more information about the sample files, see the section titled Using the Sample Files.