IMSINH in Excel: Compute Complex Hyperbolic Sines and Verify Exponentials

• Introduced: the Microsoft function catalogue marks this identity as an Excel 2013 addition. An exact first build or preview date is not established.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official IMSINH reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMSINH(inumber)

The required input uses x+yi or x+yj text. Microsoft recommends COMPLEX when the coefficients are numeric cells.

The reference explicitly documents #NUM! for a value outside the stated complex format and #VALUE! for logical input. It does not supply every blank, numeric-text or error-precedence rule.

The component equation above is an independent mathematical model of complex hyperbolic sine. The dedicated page identifies the function, syntax and a worked 4+3i example; it does not print that component equation itself.

Enter only the first column as text in A2:A5. Put =IMSINH(A2) in B2 and fill down. Extract the real and imaginary coefficients into C and D.

These are shortened independent expected displays. Keep the actual computed B values when checking identities, rather than replacing them with the printed decimals.

The last row has real coefficient zero because SINH(0)=0. Its imaginary coefficient is SIN(1), since COSH(0)=1. A purely imaginary argument therefore has a different reduction from a purely real argument.

For z=1+i, enter:

=SINH(1)*COS(1)
=COSH(1)*SIN(1)

Expected coefficients are approximately 0.634963915 and 1.298457581. A cell-driven version uses IMREAL(A2) for x and IMAGINARY(A2) for y.

SINH and COSH act on the real coefficient. Ordinary sine and cosine act on the imaginary coefficient in radians in this identity.

This is the reverse placement of ordinary and hyperbolic functions compared with the complex ordinary-sine equation. Exchanging those roles would calculate a different function.

Hyperbolic sine is half the difference between the exponentials of z and −z. For z=1+i, compute the two values with IMEXP, then subtract with IMSUB.

The exponential of 1+i has coefficients approximately 1.468693940 and 2.287355287. The exponential of −1−i has coefficients approximately 0.198766110 and −0.309559876.

Half their difference gives the IMSINH components approximately 0.634963915 and 1.298457581. A formula using explicitly constructed real division is:

=IMDIV(IMSUB(IMEXP("1+i"),IMEXP("-1-i")),COMPLEX(2,0))

This provides an independent structural check. For extremely small differences or very large real coefficients, direct exponential subtraction may lose precision or overflow; no general native equivalence at such limits is claimed.

For the real-only input 1+0i, the expected real coefficient is SINH(1), approximately 1.175201194, and the imaginary coefficient is zero.

For the imaginary-only input 0+1i, the expected real coefficient is zero and the imaginary coefficient is SIN(1), approximately 0.841470985.

The two values have the same input magnitude but different directions and different outputs. Applying ordinary SINH to only the input magnitude would discard the information needed to choose between them.

A real logarithmic growth coefficient is not automatically a degree angle. Convert angle units only where the mathematical model calls for a radian argument.

Negating both coordinates gives sinh(−z)=−sinh(z). The expected coefficients for −1−i are approximately −0.634963915 and −1.298457581.

Conjugating the input changes only its imaginary sign: 1−i has the same real output coefficient as 1+i and the opposite imaginary coefficient. These are different transformations and should not be combined casually in cleanup formulas.

A further independent identity is SIN(iz)=i×SINH(z), interpreted through the complex IMSIN and IMSINH functions. For z=1+i, iz=−1+i. Both modeled sides have coefficients approximately −1.298457581 and 0.634963915.

Microsoft’s 4+3i example has coefficients near −27.0168132580039 and 3.85373803791938. The independent component and exponential models corroborate that published control.

Logical-input #VALUE! and malformed-format #NUM! are documented rules, but their precedence in more complicated formulas is not established here. Large hyperbolic values, underflow, degenerate output TYPE and alternate string parsing remain unverified.

Use explicit component tolerances for approximate checks. No native Excel execution was performed.

• Microsoft IMSINH 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.