# How to use the IMSINH function

**What is the IMSINH function?**

The IMSINH function calculates the hyperbolic sine 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 hyperbolic sine?**

Hyperbolic functions are similar to ordinary trigonometric functions, but they use a different shape to define them.

Trigonometric functions use a circle, while hyperbolic functions use a hyperbola. The chart above shows a hyperbola and two asymptotes (dashed lines) where the intersection is at the center of the hyperbola. The chart below shows a circle containing the trigonometric functions.

**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 sine and complex hyperbolic sine?**

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

The hyperbolic sine of a real number x is defined as

sinh(x) = (e^{x} - e^{-x})/2

Natural number e is the base of the natural logarithm.

The complex hyperbolic sine of a complex number z = x + yi is defined as sinh(z) = sinh(x)cos(y) + i cosh(x)sin(y)

Complex numbers has i as the imaginary unit.

### Table of Contents

## 1. IMSINH Function Syntax

IMSINH(*inumber*)

## 2. IMSINH Function Arguments

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

## 3. IMSINH function example

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

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+i is the light blue line in the first quadrant. The hyperbolic sine of 2+i is the green line also shown in the first quadrant.

### 3.1 Explaining formula

#### Step 1 - Populate arguments

IMSINH(*inumber*)

becomes

IMSINH(B3)

#### Step 2 - Evaluate the IMSINH function

IMSINH(B3)

becomes

IMSINH("1+2i")

and returns

-0.489056259041294+1.40311925062204i

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

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

C = x + yi

sinh(x + yi) = sinh x*cos y + icosh x*sin y

For example, C=2+i

sinh(2 + i) = sinh 2*cos 1 + icosh 2*sin 1

becomes

sinh(2 + i) = 3.62686040784702*0.54030230586814 + 3.76219569108363*0.841470984807897i

equals

sinh(2 + i) = 1.95960104142161+3.16577851321617i

## 5. IMSINH function not working #NUM error

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

### Useful links

IMSINH function - Microsoft

Hyperbolic Sine of Complex Number

Hyperbolic Functions

### Functions in 'Engineering' category

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

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