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)
)
Related functions
IMCOS Function
The Excel IMCOS function returns the cosine of a complex number.
COMPLEX Function
The Excel COMPLEX function creates a complex number with given real and imaginary coefficients. COMPLEX returns a text result in the form x + yi or x + yj....
IMREAL Function
IMAGINARY Function
SIN Function
The Excel SIN function returns the sine of an angle given in radians. To supply an angle to SIN in degrees, multiply the angle by PI()/180 or use the RADIANS function to convert to radians.
