How to Use the SINH Function in Excel

Understanding the Hyperbolic Sine Function in Spreadsheets

The SINH function is a mathematical function used in both Microsoft Excel and Google Sheets. This function calculates the hyperbolic sine of a given number as its input. The hyperbolic sine function is widely used in various fields including engineering, physics, and mathematics to solve various hyperbolic and exponential equations.

Syntax and Examples

The syntax for the SINH function in both Excel and Google Sheets is straightforward:

=SINH(number) 

Where number is the value (angle in radians) for which the hyperbolic sine is calculated.

Example: To find the hyperbolic sine of 1 in a spreadsheet, you can enter:

=SINH(1) 

This function will return approximately 1.175. Similarly, you could calculate the hyperbolic sine for any other numeric value by replacing “1” in the formula with the desired number.

Utilizing SINH in Practical Scenarios

While it might initially seem that the SINH function is purely academic, it can be applied in various practical scenarios including the modeling of natural phenomena and engineering calculations.

Case 1: Calculating Cable Catenaries

In civil engineering, the catenary curve describes the shape of a hanging cable or chain under its own weight when supported only at its ends. The equation for a catenary curve often involves the hyperbolic sine function.

Problem: Assume you need to estimate the mid-point displacement of a cable that hangs between two poles 50 meters apart, with a maximum sag of 10 meters.

Using the catenary formula:

y = a * (SINH(x/a) - SINH(0)) 

Where x is the horizontal distance from the lowest point of the cable, and a can be calculated using various methodologies based on tension and length. Let’s assume in this simplified example a is 20 meters.

The formula in Excel or Google Sheets for calculating the height y at 25 meters (half the total length) from the lowest point (assuming the lowest point x = 0 is at the middle) would be:

=20 * (SINH(25/20) - SINH(0)) 

By inputting the formula, you can calculate the displacement quickly in either software.

Case 2: Modeling Hyperbolic Growth

Another use case is modeling scenarios where growth accelerates over time in a hyperbolic manner, a pattern observed in certain economic models.

Problem: Suppose an investor predicts the valuation of an innovative startup will grow hyperbolically due to progressive technology adoption. The model to calculate valuation at each year is based on the formula:

V = P * SINH(r * t) 

Where P is the initial valuation, r is the growth factor and t is time in years.

If the initial valuation (P) is $1 million, and the yearly growth factor (r) is 0.3, the formula to calculate the valuation after 3 years would be:

=1000000 * SINH(0.3 * 3) 

This calculation would provide an estimate based on the investor’s hyperbolic growth model.

Conclusion

The SINH function’s utility in Excel and Google Sheets taps into a variety of practical scenarios, extending beyond just pure mathematics into real-world applications. Whether it’s in engineering calculations for physical structuring like cable catenaries or in economic models predicting hyperbolic growth, the SINH function offers powerful solutions for sophisticated modeling. Understanding how to effectively apply this function allows users to harness the full potential of their spreadsheet tools.

More information: https://support.microsoft.com/en-us/office/sinh-function-1e4e8b9f-2b65-43fc-ab8a-0a37f4081fa7