IMSIN: Sine of a Complex Number in WPS Spreadsheet
Calculate complex sine with IMSIN using real trigonometric and hyperbolic factors. Check pure-imaginary and conjugate inputs through independent coefficients.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
IMSIN calculates the sine of a complex number. Its imaginary part changes the calculation through hyperbolic functions, so this is not obtained by applying ordinary SIN separately to both coefficients. Unlike real sine, a complex sine need not have magnitude at most one.
Syntax and mathematical units
=IMSIN(inumber)
The required argument is complex text in the x+yi or x+yj notation described by the WPS lesson, or a reference containing that value. The official formula image gives:
sin(x+yi) = sin(x)cosh(y) + cos(x)sinh(y)i
The ordinary trigonometric factors use radians. This is the mathematical identity underlying the controls; no degree-mode toggle or automatic degree conversion is established by the lesson. If the real component represents an angle measured in degrees, conversion is a separate step before constructing the input.
Calculate a genuinely complex example
For =IMSIN("1+1i"), use x=1 and y=1. The independently calculated real coefficient is sin(1)cosh(1), approximately 1.29845758142. The imaginary coefficient is cos(1)sinh(1), approximately 0.634963914785. These values specify the mathematical result, not an exact runtime string.
| Formula | Expected real coefficient | Expected imaginary coefficient |
|---|---|---|
=IMSIN("0+0i") |
0 | 0 |
=IMSIN("1+0i") |
0.841470984808 | 0 |
=IMSIN("0+1i") |
0 | 1.17520119364 |
=IMSIN("0-1i") |
0 | -1.17520119364 |
=IMSIN("1+1i") |
1.29845758142 | 0.634963914785 |
=IMSIN("1-1i") |
1.29845758142 | -0.634963914785 |
At y=0, cosh(0)=1 and sinh(0)=0, so IMSIN on a real input reduces mathematically to ordinary sine with zero imaginary coefficient. At x=0, the real coefficient is zero and the imaginary coefficient is sinh(y). Thus sin(i) has imaginary coefficient approximately 1.17520119364.
Practical check: conjugate input symmetry
The inputs 1+i and 1−i have the same real coefficient and opposite imaginary coefficients. Because cosh is even and sinh is odd, their sine results have the same real coefficient and opposite imaginary coefficients. This offers a useful manual consistency check for a formula model using conjugate pairs.
It also shows why taking sine of the two components separately is wrong: sin(1)+sin(1)i would miss both the hyperbolic scaling and the cosine factor. For a purely imaginary input, a real-sine range check would reject mathematically valid results larger than one in magnitude.
Worksheet setup and limits
You can place text 1+1i in A2 and use =IMSIN(A2) to mirror the lesson’s reference workflow. Keep coefficients, suffix and punctuation explicit. The examples do not establish exact formatting, whether a coefficient of one is printed, or how zero components are omitted.
The lesson does not state precise WPS errors for malformed complex text, empty references, suffix variants or extreme values. These remain unverified rather than being filled in from another product. Hyperbolic factors grow rapidly as the imaginary coefficient’s magnitude increases, so the bounded controls here do not provide overflow or precision guarantees for large values.
IMSIN and SIN serve different input domains. The real-input identity above is a mathematical special case, not proof that the functions have identical text coercion, return types or error behavior.
Availability and evidence
The official WPS lesson is dated 2019-06-19 and explicitly labels its applicability WPS Office for Windows Personal. It does not identify a numeric release. This establishes documentation by that date, not the function’s original introduction or its last working version. Current build limits and equivalence across WPS clients remain unverified. The available evidence does not establish retirement or formal deprecation.
Checked October 18, 2026. The examples in this guide are original, independently calculated controls based on the lesson. They have not been run in WPS or another spreadsheet application. No screenshot or execution result is claimed.