ASINH function

The ASINH function in Excel returns the inverse hyperbolic sine (arcsinh) of a number. This function is used to find the value of an angle whose hyperbolic sine is a given number. The result is expressed in radians.

Syntax

=ASINH(number)

Parameters

• number: The hyperbolic sine value for which you want to calculate the inverse. This can be any real number, positive or negative. Unlike the ASIN function, the ASINH function accepts values outside the range of -1 to 1.

Return Value

• The function returns the inverse hyperbolic sine of the number, expressed in radians.

How It Works

• The ASINH function is the inverse of the hyperbolic sine function, and it calculates the value whose hyperbolic sine is equal to the provided input.
• The formula for the inverse hyperbolic sine of a number $x$ is: $$\text{ASINH}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)$$ Where $\ln$ represents the natural logarithm.

Examples

Example 1: Basic Inverse Hyperbolic Sine Calculation

To calculate the inverse hyperbolic sine of 1:

=ASINH(1)

Result: 0.8814 (This is approximately $\ln(1 + \sqrt{1^2 + 1})$).

Example 2: Inverse Hyperbolic Sine of a Negative Number

You can also use ASINH with negative numbers. For example:

=ASINH(-1)

Result: -0.8814 (The inverse hyperbolic sine of -1).

Example 3: Handling Larger Numbers

The ASINH function can handle very large and very small numbers, unlike the ASIN function, which has a limited input range of -1 to 1.

=ASINH(100)

Result: 5.2983 (Inverse hyperbolic sine of 100).

Key Points

• The ASINH function accepts any real number, positive or negative.
• The result is expressed in radians. If you need the result in degrees, use the DEGREES function to convert the radians to degrees.
• This function is useful in mathematics, physics, and engineering, especially when working with hyperbolic functions.

Use Cases

• Hyperbolic Functions: In applications involving hyperbolic geometry, engineering, and physics, ASINH is often used to calculate the inverse of the hyperbolic sine.
• Mathematical Analysis: Useful for solving equations that involve hyperbolic sine values and their inverses.