# How to use the IMCOS function

**What is the IMCOS function?**

The IMCOS function calculates the cosine 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 the cosine?**

The cosine is a trigonometric function that relates to an angle Î¸ in a right triangle to the ratio of the length of the side adjacent the angle and the length of the longest side (hypotenuse) of the triangle. A right triangle has one angle that measures 90Â° or Ï€/2 radians which is approximately 1.5707963267949 radians.

**What is the difference between sine and the complex sine?**

The difference between cosine and complex cosine is that the former is defined for real numbers only, while the latter is defined for complex numbers as well. The cosine of a complex number has some similarities to the cosine of a real number, such as periodicity.

The cosine function is periodic considering the angle, meaning it repeats its values after a certain interval. The period of the cosine function is 2Ï€ or 360 degrees.

cosine z = (e^{iz} + e^{-iz})/2

z - complex number

i - imaginary unit

### Table of Contents

## 1. IMCOS Function Syntax

IMCOS(*inumber*)

## 2. IMCOS Function Arguments

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

## 3. IMCOS function example

The image above demonstrates a formula in cell B28 that calculates the cosine 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 B28:

### 3.1 Explaining formula

#### Step 1 - Populate arguments

IMCOS(*inumber*)

becomes

IMCOS(B25)

#### Step 2 - Evaluate IMCOS function

IMCOS(B25)

becomes

IMCOS("2+2i")

and returns

-1.56562583531574-3.29789483631124i

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

The cosine of a complex number is calculated like this:

C = x + yi

cos(C) = cos(x)*cosh(y) - sin(x)*sinh(y)i

For example, C=2+2i

cos(2+2i) = cos(2) cosh(2) - sin(2) sinh(2)i

becomes

cos(2+2i) = -0.416146836547142*3.76219569108363 + 0.909297426825682*3.62686040784702i

equals

cos(2+2i) = -1.56562583531574 - 3.29789483631124i

sin - calculates the sine of a number

cos - calculates the cosine of a number

cosh - calculates the hyperbolic cosine of a number

sinh - calculates the hyperbolic sine of a number

## 5. IMCOS function not working - #NUM error

The IMCOS function returns a #NUM error if the provided argument is not a valid complex number. The image above shows a worksheet with an invalid complex number specified in cell B25: 2+2

Excel needs an i or j in the complex number to work properly, correct the mistake and the IMCOS function will work again.

### Useful links

IMCOS function - Microsoft

The Complex Cosine and Sine Functions - Mathonline

Cosine of Complex Number - Proofwiki

### Functions in 'Engineering' category

The IMCOS function function is one of 42 functions in the 'Engineering' category.

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