Purpose
Return value
Syntax
=IMSIN(complex_num)
- complex_num - The complex number in the form “x+yi”.
Using the IMSIN function
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)
)
Purpose
Return value
Syntax
=IMSINH(complex_num)
- complex_num - The complex number in the form “x+yi”.
Using the IMSINH function
The Excel IMSINH function returns the hyperbolic sine of a complex number. For example, given 1+π/2i as input, the function returns -4.10319E-15 + 1.543080635i as output.
=IMSINH(COMPLEX(1, PI()/2)) // returns -4.10319E-15 + 1.543080635i
When the function’s output is plotted over the complex plane, the real output along the real axis traces the shape of the SINH function .
=IMSINH(COMPLEX(x,0)) // returns SINH(x) + 0i
The imaginary output along the imaginary axis traces the shape of the SIN function .
=IMSINH(COMPLEX(0,y)) // returns 0 + SIN(y)i
Explanation
The function can be defined for complex input using the SIN , COS , SINH , and COSH functions, which take real numbers as input.
