The IMCOSH function is used to return the hyperbolic cosine of a complex number input in a "x + yi" or "x + yj" format.
Contents:
Syntax
= IMCOSH(inumber)inumber = the complex number from which you want to return the hyperbolic cosine
What is a Complex Number
A complex number is a number that has both a real and an imaginary component.
Real numbers are nearly any number that you can think of, whole numbers, negatives, fractions, decimals, and even square roots:
Imaginary numbers are created when a squared number gives a negative result. Normally this cannot happen as a negative times a negative will always yield a positive value, but if we say:
When we square i in this case, we get -1. Some examples of imaginary numbers are:
Complex numbers combine both a real and and imaginary number like so:
Either part of the complex number can also be zero.
Explanation
The IMCOSH function is part of the "Engineering" group of functions within Excel.
Unless you are taking the hyperbolic cosine of a purely real number, the inumber argument must be in a text format. Any complex number must be in the form "x + yi" or "x + yj", where x represents the real number, and y represents the imaginary number. If it is not, a #NAME? error will be thrown.
The COMPLEX function can be used in conjunction with this function to aid in the creation of complex numbers.
This means that the complex number either needs to be surrounded it quotes, or a reference to a cell containing a complex number in text form.
The IMCOSH function uses the below formula to calculate the hyperbolic cosine of a complex number:
Notes:
- If either the inumber argument is not a complex number, a #NAME! error will be returned.
- The suffix of the complex number must be an "i" or a "j", if it is not, a #NUM! error will be returned.
Examples
1. How to Calculate the Hyperbolic Cosine of a Complex Number (Hardcoded)
This is the basic use case for the IMCOSH function. By hard coding in complex numbers, we can return the hyperbolic cosine.
It is important to note that the complex number must be surrounded by quotes. If you forget the quotes, Excel will try to auto correct your formula to something incorrect or you will get a #NAME! error returned.
Also note how both "i" and "j" work and return the same results.
= IMCOSH("-8+2i")
2. How to Calculate the Hyperbolic Cosine of a Complex Number (Cell References)
Similar to the example above, the IMCOSH function can also be fed text inputs as cell references. When referencing text in this way, you do not need to include the double quotes around the imaginary suffice (i or j), as the function will interpret it automatically. Check out the COMPLEX function to create complex numbers much easier.
= IMCOSH(B3)
3. How to Calculate the Hyperbolic Cosine of Imaginary Numbers
Using the IMCOSH function can also be a good way of calculating the hyperbolic cosine of imaginary numbers if you know the coefficients needed, without having to manually type them out. All you need to do is reference a set of imaginary numbers and the formula will do the rest.
Note how both "i" and "j" work and return the same results:
= IMCOSH(B3)