IMSIN in Excel: Calculate Complex Sines and Audit Hyperbolic Components

• Introduced: the first introduction version is not established by the reviewed official history.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2019 for Mac, Excel 2016.
• Checked: October 14, 2026; official IMSIN reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMSIN(inumber)

Inumber is required and uses complex i or j notation. COMPLEX constructs it from numeric real and imaginary coefficients.

The official equation image gives real coefficient SIN(x)×COSH(y) and imaginary coefficient COS(x)×SINH(y). Ordinary trigonometric arguments use radians. The imaginary coefficient is not handled by applying ordinary SIN to y independently.

The dedicated page supplies this equation and an explicit worked complex example, but does not provide a complete malformed-input or coercion/error table. These fixtures use explicit valid complex values.

Enter only the first column as text in A2:A5. In B2 use =IMSIN(A2) and fill down. In C and D extract real and imaginary output coefficients.

The third row reverses only the imaginary input coefficient relative to the second. Its sine is the conjugate of the second result in the independent component model.

The final row has zero real coordinate, so SIN(0)=0 and COS(0)=1. Its imaginary output coefficient is SINH(1), approximately 1.175201194.

Use =IMREAL(B2) and =IMAGINARY(B2) to obtain numeric values. The table shows shortened expected displays, not full native complex-string captures.

For the first row, calculate:

=SIN(0.5)*COSH(0.25)
=COS(0.5)*SINH(0.25)

The independently expected values are approximately 0.494485781 and 0.221688164. Constructing them with COMPLEX gives the same modeled complex sine.

For a cell-driven check, replace the constants with IMREAL(A2) and IMAGINARY(A2). The sign in the second coefficient is positive in the sine equation; the corresponding complex cosine equation has a different sign.

COSH and SINH are real hyperbolic functions used on numeric helper values. They are not substitutes for the whole complex operation.

When the imaginary input coefficient is zero, the real result reduces to ordinary SIN(x), while the imaginary result is zero mathematically.

For an angle of 30 degrees, use =IMSIN(COMPLEX(RADIANS(30),0)). The expected real coefficient is 0.5 and the imaginary coefficient is zero.

Entering 30 directly as the real coefficient would mean 30 radians. Keep source units explicit; a cell formatted with a degree symbol does not automatically convert its numeric value to radians.

These simplified zero-component quantities do not certify every native output TYPE or the exact text used for a tiny floating residual.

For z=1+i, the real sine coefficient is approximately 1.298457581, already larger than one. This is not a contradiction: the familiar interval from −1 to one applies to sine of real arguments.

The complex result includes the hyperbolic factors in the documented equation. Increasing the size of the imaginary input can increase the magnitude substantially.

Do not clamp complex sine components to the real sine interval as a cleanup step. That would change a valid mathematical result and could break later identities.

For z=1+i, compute s with IMSIN and c with IMCOS. Square the complete complex quantities and add them:

=IMSUM(IMPRODUCT(IMSIN("1+i"),IMSIN("1+i")),IMPRODUCT(IMCOS("1+i"),IMCOS("1+i")))

The mathematical target has real coefficient one and imaginary coefficient zero. This is an identity involving complex squares, not squares of magnitudes.

Replacing each square with IMABS of the value squared answers a different question. For this input, the two squared magnitudes sum to COSH(2), approximately 3.762195691, rather than one.

Sine is odd in the independent mathematical model: reversing both input coefficients reverses both output coefficients. The result for −1−i therefore has expected coefficients approximately −1.298457581 and −0.634963915.

Microsoft’s separate 4+3i example has expected coefficients near −7.61923172032141 and −6.548120040911, consistent with the documented equation. This source control does not establish every unlisted input form or error.

Use explicit tolerances for component checks. Native overflow, underflow, malformed-string errors and extreme precision remain unverified; no native Excel execution was performed.

• Microsoft IMSIN reference.
• Editorial testing standards.

Expected outputs are independently calculated examples or documented rules, not captured native Excel results. Formulas use English function names and comma argument separators. Locale, rendering and boundary restrictions are stated where relevant.