How to use the IMCOSH function
What is the IMCOSH function?
The IMCOSH function calculates the hyperbolic cosine of a complex number in x + yi or x + yj text format.
The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.
What is the hyperbolic cosine?
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 a hyperbola. The chart above shows a hyperbola and two asymptotes (dashed lines) where the intersection is at the center of the hyperbola. The chart below shows a circle containing the trigonometric functions.
What is a hyperbola?
The equation of a hyperbola with a horizontal axis is
(x2/ a2) - (y2Â / b2) = 1
where a and b are positive constants.
A circle has a constant distance from the center point, while a hyperbola is a curve that has two focus points (+ae, 0), and (-ae, 0).
What is the difference between hyperbolic cosine and complex hyperbolic cosine?
The difference between hyperbolic cosine and hyperbolic cosine for complex numbers is that the former is defined for real numbers, while the latter is defined for complex numbers.
The hyperbolic cosine of a real number x is defined as
cosh(x) = (ex + e-x)/2
Natural number e is the base of the natural logarithm.
The complex hyperbolic cosine of a complex number z = x + yi is defined as cosh(z) = cosh(x)cos(y) + i sinh(x)sin(y)
Complex numbers has i as the imaginary unit.
Table of Contents
1. IMCOSH Function Syntax
IMCOSH(inumber)
2. IMCOSH Function Arguments
| inumber | Required. A complex number in x+yi or x+yj text format. |
3. IMCOSH function example
The image above demonstrates a formula in cell B28 that calculates the hyperbolic cosine of a complex number specified in cell B25.
Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.
The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.
Formula in cell B28:
The chart above shows the complex plane, the y-axis is the imaginary axis and the x-axis is the real axis.
Complex number 2+i is the light blue line in the first quadrant. The hyperbolic cosine of 2+i is the green line also shown in the first quadrant.
3.1 Explaining formula
Step 1 - Populate arguments
IMCOSH(inumber)
becomes
IMCOSH(B25)
Step 2 - Evaluate the IMCOSH function
IMCOSH(B25)
becomes
IMCOSH("2+i")
and returns
2.03272300701967+3.0518977991518i
4. How the IMCOSH function is calculated in detail
The hyperbolic cosine of a complex number is calculated like this:
C = x + yi
cosh(x + yi) = cosh(x)*cos(y) + isinh(x)*sin(y)
For example, C=2+i
cosh(2 + i) = cosh(2)*cos(1) + isinh(2)*sin(1)
becomes
cosh(2 + i) = 3.76219569108363*0.54030230586814 + 3.62686040784702*0.841470984807897i
equals
cosh(2 + i) = 2.03272300701967+3.0518977991518i
5. IMCOSH function not working
The IMCOSH function returns a #NUM error if the provided argument is not a valid complex number.
Useful links
IMCOSH function - Microsoft
Hyperbolic functions for complex numbers
Hyperbolic functions – Graphs, Properties, and Examples
Hyperbolic curve fitting in Excel
Functions in 'Engineering' category
The IMCOSH function function is one of 42 functions in the 'Engineering' category.
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