SINH function

The SINH function in Excel calculates the hyperbolic sine of a number. The hyperbolic sine is defined using the formula:

$$\text{sinh}(x) = \frac{e^x – e^{-x}}{2}$$

Where $e$ is the base of the natural logarithm (approximately 2.718), and $x$ is the input number.

Syntax

=SINH(number)

Parameters

• number (required): The number for which you want to calculate the hyperbolic sine.

Key Points

1. Hyperbolic Functions: The hyperbolic sine function is similar to the regular sine function, but it applies to hyperbolic geometry and is used in certain types of mathematical and physical modeling.
2. Output Range: The result of the SINH function can be any real number, depending on the input.
3. Application: The hyperbolic sine function is often used in the study of waveforms, fluid dynamics, electrical engineering, and hyperbolic geometry.

Examples

1. Calculate the hyperbolic sine of 1:

   =SINH(1)
   

   Result: 1.175201 (since $\sinh(1) \approx 1.1752$)

2. Calculate the hyperbolic sine of 0:

   =SINH(0)
   

   Result: 0 (since $\sinh(0) = 0$)

3. Calculate the hyperbolic sine of -1:

   =SINH(-1)
   

   Result: -1.175201 (since $\sinh(-1) \approx -1.1752$)

4. Calculate the hyperbolic sine for a number in cell A1:

   =SINH(A1)
   

Notes

• Behavior for Large Numbers: The SINH function can return very large values for large input numbers due to the exponential nature of the formula.
• Negative Inputs: The hyperbolic sine is an odd function, meaning that $\sinh(-x) = -\sinh(x)$.

Related Functions

• COSH: Calculates the hyperbolic cosine, another hyperbolic function often used in conjunction with SINH.
• TANH: Calculates the hyperbolic tangent, which is the ratio of the hyperbolic sine and cosine.
• ASINH: The inverse hyperbolic sine function, returns the value whose hyperbolic sine is a given number.
• EXP: Returns the value of $e^x$, which is part of the formula for the hyperbolic sine.

The SINH function is valuable in advanced mathematical modeling, particularly in fields that deal with exponential growth, waveforms, and hyperbolic trigonometry.