## How to use the IMCSC function

The IMCSC function calculates the cosecant of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

**What is a complex number?**

A complex number contains a real and imaginary value, they let you for example solve equations with no real solutions like x^{2} + 1 = 0Â The chart above shows x^{2} + 1 = 0 and it never touches the x-axis.

However, mathematicians invented the imaginary number and named it i, it extends into the complex plane and lets you solve equations using imaginary numbers.

**What is the cosecant?**

The cosecant is one of the six trigonometric functions, it is the multiplicative inverse of the sine function. This means that it is equal to 1 divided by the sine of a given angle Î¸.

The cosecant is csc Î¸ = 1 / sin Î¸. In a right-angled triangle, the cosecant of an angle Î¸ is equal to the hypotenuse divided by the opposite side.

**What is the complex cosecant?**

The complex cosecant is the extension of the cosecant function to the complex plane. It is defined as the reciprocal or the multiplicative inverse of the complex sine function, which means that it is equal to 1 divided by the sine of a complex number.

csc z = 1 / sin z, where z is a complex number.

The complex sine function can be expressed using the ordinary sine and cosine functions and the hyperbolic functions.

sin (x+yi) = sin x cosh y + i cos x sinh y

### Table of Contents

## 1. IMCSC Function Syntax

IMCSC(*inumber*)

## 2. IMCSC Function Arguments

inumber |
Required. A complex number in x+yi or x+yj text format. |

## 3. IMCSC function example

The image above demonstrates a formula in cell B28 that calculates the cosecant of a complex number specified in cell B25.

Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.

The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.

Formula in cell D3:

The chart above shows the complex plane, the y-axis is the imaginary axis and the x-axis is the real axis.

Complex number 2+2i is the light blue line in the first quadrant. The cosecant of 2+2i is the small green line also displayed in the first quadrant.

### 3.1 Explaining formula

#### Step 1 - Populate arguments

IMCSC(*inumber*)

becomes

IMCSC(B25)

#### Step 2 - Evaluate the IMCSC function

IMCSC(B25)

becomes

IMCSC("2+2i")

and returns

0.244687073586957+0.107954592221385i

## 4. How is the IMCSC function calculated in detail?

The cosecant of a complex number is calculated like this:

C = x + yi

csc(C) = (sin(x)*cosh(y) - icos(x)*sinh(y)) / (sinÂ²(x)*coshÂ²(y) + icosÂ²(x)*sinhÂ²(y))

For example, C = 2 + 2i

csc(2+2i) = (sin(2)*cosh(2) - icos(2)*sinh(2)) / (sinÂ²(2)*coshÂ²(2) + icosÂ²(2)*sinhÂ²(2))

becomes

csc(2+2i) = (3.42095486111701 - (-1.50930648532362i)) / 13.9809382284401

equals

csc(2+2i) = 0.244687073586957+0.107954592221385i

## 5. IMCSC function not working

The IMCSC function returns a #VALUE! error if the argument is a boolean value.

The IMCSC function returns a #NUM! error if the argument is an invalid complex number. The i is missing in cell B25 shown in the image above.

### Useful links

IMCSC function - Microsoft

Cosecant of Complex Number

Trigonometric functions - Wikipedia

### Functions in 'Engineering' category

The IMCSC function function is one of many functions in the 'Engineering' category.

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