## Purpose

## Return value

## Syntax

`=IMSIN(complex_num)`

*complex_num*- The complex number in the form "x+yi".

## How to use

The Excel IMSIN function returns the sine of a complex number. For instance, given "1+1i" as input, the function returns the complex number equal to the sine of the input.

`=IMSIN("1+1i") // returns 1.29845758141598+0.634963914784736i`

Given real number input, the function behaves like the sine function. For example, given π + 0i as input the function returns 3.23108914886517E-15 (approximately zero). The sine of π is zero, but due to floating-point precision, it returns a very small number close to zero.

`=IMSIN(COMPLEX(PI(),0)) // returns 3.23108914886517E-15`

### Explanation

Mathematically, the sine of a complex number can be represented using a combination of the standard and hyperbolic trigonometric functions.

If B6 contains a complex number in the form "x+yi", this is equivalent to the following formula.

```
=COMPLEX(
SIN(IMREAL(B6))*COSH(IMAGINARY(B6)),
COS(IMREAL(B6))*SINH(IMAGINARY(B6))
)
```

Alternatively, the sine of a complex number can be defined in terms of the exponential function, where "z=x+yi".

If B6 contains a complex number in the form "x+yi", this is equivalent to the following formula.

```
=IMDIV(
IMSUB(
IMEXP(IMPRODUCT(COMPLEX(0,1), B6)),
IMEXP(IMPRODUCT(COMPLEX(0,-1), B6))
),
COMPLEX(0, 2)
)
```