A to Z of Excel Functions: The IMSINH Function
21 December 2020
Welcome back to our regular A to Z of Excel Functions blog. Today we look at the IMSINH function.
The IMSINH 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.
![](http://sumproduct-4634.kxcdn.com/assets/blog-pictures/2020/a-to-z/236/image1.gif)
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,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f1.1.png/c2252580a5d243da597e57f2abba8508.jpg)
By using the basic trigonometric ratios,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f2.png/a62f6fb677ff4000d756979eb0ebdcd2.jpg)
Therefore, multiplying each side by r:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f3.png/20989b8738b96b49fae50c4825b1e49c.jpg)
Therefore,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f4.png/4b16297c45de4f462633df9cd7be2a2d.jpg)
In the case of a complex number, r represents the absolute value, or modulus,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f5.png/e26cb93271653fb859f28e0bc673668c.jpg)
and the angle θ is called the argument of the complex number
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f66.png/a615df6aa0461efb011a2744b229e964.jpg)
Using Euler’s Formula,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/image2.png/f32e5a15e2cf9c3e4d2d058458ce054d.jpg)
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f7.png/c7a317caba746d90f323ed52c4a4e374.jpg)
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/image3.png/f1140ff857fc3b6f5f97a6a24f4a6fc7.jpg)
you eventually get:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f8.png/5f25b98dddde956401fc5dafd246972d.jpg)
The IMSINH function returns the hyperbolic sine of a complex number in x + yi or x + yj text format.
The IMSINH function employs the following syntax to operate:
IMSINH(inumber)
The IMSINH function has the following argument:
- inumber: this is required and represents the complex number for which you want to calculate the hyperbolic sine.
It should be further noted that:
- you should use COMPLEX to convert real and imaginary coefficients into a complex number
- IMSINH recognises either the i or j notation
- if inumber is a value that is not in the x + yi or x + yj text format, IMSINH returns the #NUM! error value
- if inumber is a logical value, IMSINH 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 IMSINH will return an #NUM! error.
Please see my example below:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/image4.png/72aa864d2854c6fefb1083fba0ab5792.jpg)
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.