A to Z of Excel Functions: The IMCOSH Function
27 July 2020
Welcome back to our regular A to Z of Excel Functions blog. Today we look at the IMCOSH function.
The IMCOSH function
An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i (sometimes denoted j) which is defined by its property i2 = −1. In general, the square of an imaginary number bi is −b2. For example, 9i is an imaginary number, and its square is −81. Zero is considered to be both real and imaginary.
An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.
The polar form of a complex number is another way to represent the number. The form z = a + bi is called the rectangular form of a complex number.
The horizontal axis is the real axis and the vertical axis is the imaginary axis. You can find the real and imaginary components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis.
From the Pythagorean Theorem,
r2 = a2 + b2
By using the basic trigonometric ratios,
cos θ = a / r and sin θ = b / r
Therefore, multiplying each side by r:
r cos θ = a and r sin θ = b
z = a + bi
z = r cos θ + (r sin θ)i
z = r(cos θ + i sin θ)
In the case of a complex number, r represents the absolute value, or modulus (where r = |z| = ), and the angle θ is called the argument of the complex number ( for a > 0 and for a < 0).
Using Euler’s Formula,
It then follows
you eventually get:
cosh(x + yi) = cosh x cos y – i sinh x sin y
The IMCOSH function returns the hyperbolic cosine of a complex number in x + yi or x + yj text format.
The IMCOSH function employs the following syntax to operate:
The IMCOSH function has the following argument:
- inumber: this is required and represents the complex number for which you want to calculate the hyperbolic cosine.
It should be further noted that:
- you should use COMPLEX to convert real and imaginary coefficients into a complex number
- IMCOSH recognises either the i or j notation
- if inumber is a value that is not in the x + yi or x + yj text format, IMCOSH returns the #NUM! error value
- if inumber is a logical value, IMCOSH returns the #VALUE! error value
- if the complex number ends in +i or -i (or j), i.e. there is no coefficient between the operator and the imaginary unit, there must be no space, otherwise IMCOSH will return an #NUM! error.
Please see my example below:
We’ll continue our A to Z of Excel Functions soon. Keep checking back – there’s a new blog post every business day.
A full page of the function articles can be found here.