How to use the ATANH function
What is the ATANH function?
The ATANH function calculates theĀ inverse hyperbolic tangent of a number.
What is hyperbolic?
Hyperbolic functions are similar to ordinary trigonometric functions, but they use a different shape to define them.
Trigonometric functions use a circle, while hyperbolic functions use hyperbolas. The chart above shows a hyperbola and two asymptotes (dashed lines) where the intersection is at the center of the hyperbola.
The unit hyperbola looks like this:
It shows x2 -y2=1 and x2 -y2=-1
What is the hyperbolic tangent?
The hyperbolic tangent is defined as
tanh(x) = sinh(x) / cosh(x)
or
tanh(x) = (e^x - e^-x) / (e^x + e^-x)
What is the inverse hyperbolic tangent?
The inverse hyperbolic tangent is defined as atanh(x) = 0.5 * log((1 + x) / (1 - x))
It is often displayed in scientific calculators as tanh-1. The atanh is related to tanh: tanh(atanh(x)) = x
What are asymptotes?
Asymptotes are straight lines that the function curves towards but never intersects.
The inverse hyperbolic tangent (atanh) has vertical asymptotes at x = Ā±1.
What is the domain of the inverse hyperbolic tangent?
The domain is -1 < x < 1
What is the range of the inverse hyperbolic tangent?
The range is all real numbers.
What are the domain, range and asymptotes of all the main hyperbolic functions?
This table displays the key properties of the main hyperbolic functions:
Hyperbolic Function | Domain | Asymptotes | Range |
sinh(x) | All real numbers | Vertical asymptotes at multiples of Ā±Ļi/2 | (-ā, ā) |
cosh(x) | All real numbers | Horizontal asymptotes at y=0 | (1, ā) |
tanh(x) | All real numbers | Horizontal asymptotes at y=Ā±1 | (-1, 1) |
sech(x) | All real numbers | Horizontal asymptotes at y=0 | (0, 1) |
csch(x) | All real x ā 0 | Vertical asymptotes at x=0 | (0, ā) |
coth(x) | All real numbers | Horizontal asymptotes at y=Ā±1 | (1, ā) |
The asymptotes shows the exponential nature and infinite limits of hyperbolic functions.
What are the domain, range and asymptotes of all the main inverse hyperbolic functions?
This table displays the key properties of the inverse hyperbolic functions:
Inverse Hyperbolic Function | Domain | Asymptotes | Range |
arcsinh(x) | All real numbers | Horizontal asymptote at y=0 | (-ā, ā) |
arccosh(x) | x>=1 | None | (0, ā) |
arctanh(x) | -1<x<1 | Vertical asymptotes at x=Ā±1 | (-ā, ā) |
arcsech(x) | 0<x<1 | None | (0, ā) |
arccsch(x) | All real x ā 0 | Vertical asymptote at x=0 | (-ā, ā) |
arccoth(x) | All real numbers | None | (0, ā) |
Excel Function Syntax
ATANH(number)
Arguments
number | Required. Must be between -1 and 1. -1 < number < 1. |
Comments
=ATANH(B3)
ATANH(TANH(number)) =Ā number.
Use the DEGREES function to convert radians to degrees.
Functions in 'Math and trigonometry' category
The ATANH function function is one of 73 functions in the 'Math and trigonometry' category.
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