# How to use the ASINH function

**What is the ASINH function?**

The ASINH function calculates the inverse hyperbolic sine of a number.

**What is the SINE function?**

The sine is a trigonometric function that relates an angle Î¸ in a right triangle to the ratio of the length of the side opposite the angle and the length of the longest side (hypotenuse) of the triangle. A right triangle has one angle that measures 90Â° or Ï€/2 radians which is approximately 1.5707963267949 radians.

The SIN function calculates the ratio between the opposite side and the hypotenuse.

**What is the hyperbolic sine?**

The hyperbolic sine, abbreviated as sinh, is one of the main hyperbolic functions.

Hyperbolic sine is defined as:

sinh(x) = (e^x - e^-x) / 2

**What is the difference between the hyperbolic sine and the trigonometric sine?**

The differences between sinh and the regular trigonometric sine is that regular sine is defined in terms of angles on the unit circle while the sinh function is defined in terms of exponential functions. Sine relates to circles, sinh relates to hyperbolas.

Sine is defined in terms of angles on the unit circle and is periodic repeating every 2Ï€ radians. The hyperbolic sine is not repeating. Sine takes an angle as input while the sinh takes a real number. Sine has values between -1 and 1 in contrast to the sinh which is undefined.

**What is hyperbolic/hyperbolas?**

Hyperbolic functions are similar to ordinary trigonometric functions, but they use a different shape to define them called 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 x^{2}Â -y^{2}=1 and x^{2}Â -y^{2}=-1

**What is the inverse hyperbolic sine?**

It calculates the hyperbolic value x which would be the angle Î¸ in terms of trigonometric functions. However, the hyperbolic value is not an angle but a real number.

arcsinh(x) = ln(x + âˆš(x2 + 1))

**What are the asymptotes for the inverse hyperbolic sine?**

Asymptotes refer to a straight lines that the function approaches but never reaches. The inverse hyperbolic sine function, asinh(x), has a horizontal asymptote at y = 0

**What is LN?**

LN is the natural logarithm, e is the base to the natural logarithm.

### Excel Function Syntax

ASINH(*number*)

### Arguments

number |
Required. Any real number. |

=ASINH(B3)

### Comments

The inverse hyperbolic sine is the value whose hyperbolic sine isÂ *number*, so ASINH(SINH(number)) equalsÂ *number*.

### Useful links

ASINH function - Microsoft

Graphs of Hyperbolic Functions

### Functions in 'Math and trigonometry' category

The ASINH function function is one of 73 functions in the 'Math and trigonometry' category.

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