# 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 **i ^{2}** = −1. In general, the square of an imaginary number

**bi**is

**−b**. For example, 9

^{2}**i**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 **rea**l 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,

**r ^{2} = a^{2} + b^{2}**

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**

Therefore,

**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**,

Given

then

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:

**IMCOSH(inumber)**

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. *