A to Z of Excel Functions: The IMSIN Function
14 December 2020
Welcome back to our regular A to Z of Excel Functions blog. Today we look at the IMSIN function.
The IMSIN 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/231/image2.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/231/formula1.png/5f07840847396f1a4166b7de5c2dd029.jpg)
By using the basic trigonometric ratios,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/231/formula2.png/37938e4090258439f1b681de3524a582.jpg)
Therefore, multiplying each side by r:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/231/formula3.png/29294e1d316d7e916c8a2f1727a688ff.jpg)
Therefore,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/231/formula4.png/0e3654c8cc77d6d9adb33a668919a1b6.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/231/formula5.png/fa2b8b34f92216876d0b9a6d5bc034da.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/231/formula6.png/c3cf061d9737fba9bb6d4c30efc6631c.jpg)
Using Euler’s Formula,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/232/image2.png/f32e5a15e2cf9c3e4d2d058458ce054d.jpg)
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/232/formula3.png/29294e1d316d7e916c8a2f1727a688ff.jpg)
you eventually get:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/232/formula2.png/37938e4090258439f1b681de3524a582.jpg)
The IMSIN function returns the sine of a complex number in x + yi or x + yj text format.
The IMSIN function employs the following syntax to operate:
IMSIN(inumber)
The IMSIN function has the following argument:
- inumber: this is required and represents the complex number for which you want to calculate the sine.
It should be further noted that:
- you should use >COMPLEX to convert real and imaginary coefficients into a complex number
- IMSIN recognises either the i or j notation
- if inumber is a value that is not in the x + yi or x + yj text format, IMSIN returns the #NUM! error value
- if inumber is a logical value, IMSIN 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 IMSIN will return an #NUM! error.
Please see my example below:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/232/image5.png/36776d1da4d05b45bb5a5d09375f407c.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.