ATANH function

The ATANH function in Excel returns the inverse hyperbolic tangent, or the arctanh, of a number. It is used to find the angle whose hyperbolic tangent (tanh) is a given value. The inverse hyperbolic tangent function is the reverse of the TANH function.

Syntax

=ATANH(number)

Parameters

• number: A numeric value representing the hyperbolic tangent (tanh) for which you want to find the inverse. The value must be between -1 and 1, exclusive. That is, the number must be greater than -1 and less than 1 (-1 < number < 1).

Return Value

The function returns the inverse hyperbolic tangent of the number, expressed in radians. The result is in the range from -∞ to +∞.

How It Works

The ATANH function calculates the inverse of the hyperbolic tangent (tanh). Hyperbolic functions like tanh, sinh, and cosh are used to describe certain types of curves and are widely used in engineering, physics, and other fields.

The relationship between tanh(x) and ATANH(x) can be expressed as:

• If y = ATANH(x), then tanh(y) = x

Example 1: Basic Calculation

To find the inverse hyperbolic tangent of 0.5:

=ATANH(0.5)

Result: 0.5493 radians (This is approximately 31.9°). This is the angle whose hyperbolic tangent is 0.5.

Example 2: Negative Value

For the inverse hyperbolic tangent of -0.5:

=ATANH(-0.5)

Result: -0.5493 radians (approximately -31.9°). This is the angle whose hyperbolic tangent is -0.5.

Example 3: Limitations

If you enter a number that is less than -1 or greater than 1, Excel will return an error because the inverse hyperbolic tangent is undefined outside of this range.

For example, using 1.5:

=ATANH(1.5)

Result: #NUM! (Excel returns an error because 1.5 is outside the valid range of -1 to 1).

Key Points

• The input to the ATANH function must be between -1 and 1, exclusive. If the number is outside this range, the function will return an error.
• The result is given in radians. If you need the result in degrees, you can use the DEGREES function to convert it:

=DEGREES(ATANH(0.5))

This converts the result to degrees.

Use Cases

• Mathematics and Engineering: In certain engineering fields, the inverse hyperbolic tangent function is used in signal processing, control systems, and complex analysis.
• Physics: It can be used in the context of calculating angles in certain types of motion or when working with hyperbolic geometry.
• Data Analysis: The ATANH function is also useful in areas that involve transformation and scaling, such as in normalizing data or working with statistical models.