# How to use the IMSECH function

**What is the IMSINH function?**

The IMSECH function calculates the hyperbolic secant 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 secant?**

It is a trigonometric function that defines an angle of a right-angled triangle to the ratio of hypotenuse and the adjacent side. In other words, it is the inverse or reciprocal of cosine.

The secant function has a domain of all real numbers except where cos Î¸ = 0, and a range of all real numbers except the interval -1 < y < 1

cos Î¸ = 0 when Î¸ is 90 degrees or Ï€/2

**What is the hyperbolic secant?**

The hyperbolic secant is related to the hyperbolic cosine. sech x = 1/(cosh x) The hyperbolic secant has real numbers between 0 and 1.

**What is a hyperbola?**

The equation of a hyperbola with a horizontal axis is

(x^{2}/ a^{2}) - (y^{2}Â / b^{2}) = 1

where a and b are positive constants.

A circle has a constant distance from the center point, while a hyperbola is a curve that has two focus points (+ae, 0), and (-ae, 0).

**What is the difference between hyperbolic secant and complex hyperbolic secant?**

The difference between hyperbolic secant and hyperbolic secant for complex numbers is that the former is defined for real numbers, while the latter is defined for complex numbers.

The hyperbolic secant of a real number x is defined as sech(x) = 2/(e^{x} + e^{-x})

Natural number e is the base of the natural logarithm.

The complex hyperbolic secant of a complex number z = x + yi is defined as IMSECH(C) = (cosh x cos y - isinh x sin y) / (cosh^{2} x cos^{2} y+ sinh^{2} x sin^{2} y)

Complex numbers has i as the imaginary unit.

### Table of Contents

## 1. IMSECH Function Syntax

IMSECH(*inumber*)

## 2. IMSECH Function Arguments

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

## 3. IMSECH function example

The image above shows a formula in cell B28 that calculates the hyperbolic secant 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 fourth quadrant. The hyperbolic secant of 2-2i is the green line located in the second quadrant.

### 3.1 Explaining formula

#### Step 1 - Populate arguments

IMSECH(*inumber*)

becomes

IMSECH(B3)

#### Step 2 - Evaluate the IMSECH function

IMSECH(B3)

becomes

IMSECH("1+2i")

and returns

-0.41314934426694-0.687527438655479i

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

The hyperbolic secant of a complex number is calculated like this:

C = x + yi

IMSECH(C) = (cosh x cos y - isinh x sin y) / (cosh^{2} x cos^{2} y+ sinh^{2} x sin^{2} y)

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

The IMSECH function returns a #NUM error if the provided argument is not a valid complex number.

### Useful links

IMSECH function - Microsoft

Hyperbolic Secant of Complex Number

Complex trigonometric definitions

### Functions in 'Engineering' category

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

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