IMCSC in ONLYOFFICE: Complex cosecant with Examples

=IMCSC(inumber)

The vendor specifically calls this an engineering function and documents only inumber; it does not list optional parameters, a default value or a multi-argument form (IMCSC vendor reference). inumber can be prepared from two numbers using the separate officially documented COMPLEX(real_num, i_num, [suffix]) function. COMPLEX accepts lowercase "i" or "j" and defaults to "i" if its optional suffix is omitted; this default belongs to COMPLEX, not to IMCSC.

Formula conventions: ONLYOFFICE’s Insert functions manual says formulas begin with = and argument lists use commas in its documented interface. The samples below use English function names, decimal points and comma separators. The manual does not establish that every localized deployment uses identical separators or translated names; check your installed application’s regional settings. Radians are used in these mathematical trigonometric identities; no documented degrees option exists for IMCSC.

Write the input as z = a + bi, where a and b are real numbers and i²=-1. The underlying mathematical definition is

csc(z) = 1 / sin(z), wherever sin(z) is nonzero.

For a complex result u+vi, its components can be predicted independently:

• Real part: Re(1 / (sin(a) cosh(b) + i cos(a) sinh(b)))
• Imaginary coefficient: Im(1 / (sin(a) cosh(b) + i cos(a) sinh(b)))

These are mathematical identities, not extra argument syntax supplied by ONLYOFFICE. In the spreadsheet, the documented IMREAL and IMAGINARY functions can be used to inspect the numerical components returned by the original function. The rendered complex-number text, the sign of negative zero, the suffix retained in the output, and internal rounding are not guaranteed by the referenced help pages.

Useful identity: csc(-z) = -csc(z); csc(z)·sin(z) = 1 away from poles. These equalities refer to mathematical functions at defined values and are useful as sanity checks, but do not imply that floating-point worksheet values match character-for-character.

1. Open a new empty sheet in ONLYOFFICE Spreadsheet Editor. As documented by the Insert functions manual, you can type directly in the formula bar or use the Formula tab → Function dialog.

2. Set A1:H1 to Scenario, Real a, Imaginary b, Unit, Complex input, IMCSC result, Observed real, Observed imaginary. If the header IMCSC results are long, widen columns E through H.

3. In B2 enter 0, in C2 enter 0, and in D2 enter i (plain text, without quoting it in a cell). Continue with the numeric inputs in the table below; one scenario per row. Enter j in D only for rows explicitly marked j. IMCSC zero-control note: B2:C2 are both zero intentionally: csc(0) is mathematically undefined. Do not expect a particular native error in F2, G2, or H2; record that output if you perform a Desktop/web test. The first defined numerical example in the reference table starts on row 3 (B3=0.5, C3=0, D3=i). Keep row 2 as a singularity test, or delete row 2 and adjust the listed row numbers accordingly.

4. In E2, enter =COMPLEX(B2,C2,D2); this creates a complex argument in the vendor’s documented format. Copy E2 down through the scenarios. COMPLEX can also be written as =COMPLEX(B2,C2) to request its documented default i suffix.

5. In F2, enter =IMCSC(E2). Copy F2 down through the scenarios. This is the actual formula under review, not an execution claim.

6. To inspect the result numerically, enter =IMREAL(F2) in G2 and =IMAGINARY(F2) in H2, then copy them down. The table’s last two columns provide independently calculated reference components. If the application displays fewer digits, format G:H as numbers with at least six decimal places.

7. Compare your native G:H values against the calculated expectations using a reasonable absolute or relative tolerance (for example 1E-6 on these moderate examples). Record which Desktop build or Docs/web release, locale and workbook format you tested.

Worksheet-reference predictions — NOT screenshot captures or observed ONLYOFFICE results:

Expected parts are rounded to six decimal places for readability. Use a tolerance rather than requiring equality to the rounded reference values. Values in E and F are complex results, whereas G and H are numerical component readouts. This distinction prevents falsely asserting the application’s exact display of strings such as 1+i, 1+1i or 1+j.

Five quick formulas you can inspect on a blank worksheet

The following one-cell calls use COMPLEX to avoid hand-typing a complex literal. Their mathematical component predictions follow from the table above; display and error details remain untested.

=IMCSC(COMPLEX(0.5,0))
=IMCSC(COMPLEX(1,1))
=IMCSC(COMPLEX(0,1))
=IMCSC(COMPLEX(-1,2))
=IMCSC(COMPLEX(1,1,"j"))

For instance, =IMCSC(COMPLEX(1,1,"j")) and =IMCSC(COMPLEX(1,1,"i")) represent the same mathematical complex coordinate. Whether their displayed strings preserve j or i is a separate native output-format question, not established by the official function summary.

1. Real-angle reciprocal. For z=0.5+0i, csc(z)=1/sin(0.5)≈2.085830 with zero imaginary part.
2. Imaginary-axis result. For z=0+1i, csc(i)=−i/sinh(1), so the mathematical imaginary component is ≈−0.850918.
3. Reciprocal worksheet. At z=1+0.5i compare IMCSC(COMPLEX(B2,C2)) with IMDIV(COMPLEX(1,0),IMSIN(COMPLEX(B2,C2))).
4. Zero-input caution. The mathematical value csc(0) is undefined. A native ONLYOFFICE test is required to determine how it displays the singularity.

Reciprocal check

At regular points multiply csc(z) by sin(z) and check that the result is mathematically 1.

Imaginary axis

csc(i·b)=−i·csch(b) for real nonzero b.

Odd symmetry

csc(-z)=−csc(z) wherever defined; reverse both complex components.

Sheet formulas for an independent reciprocal or symmetry check

Place real part a in B2 and imaginary coefficient b in C2, then use these formulas in any blank cells:

=COMPLEX(B2,C2)
=COMPLEX(-B2,-C2)
=IMCSC(COMPLEX(B2,C2))
=IMCSC(COMPLEX(-B2,-C2))
=IMREAL(IMCSC(COMPLEX(B2,C2)))
=IMAGINARY(IMCSC(COMPLEX(B2,C2)))

The same worksheet permits examining the sign/parity identity above by comparing the final two results with their counterparts at -z. Comparing floating-point numeric components is generally more meaningful than directly comparing complex-number text strings.

Complex trigonometry identity using related functions

For a non-singular input, the following are mathematically equivalent. They are distinct test proposals that have not been run inside ONLYOFFICE:

=IMDIV(COMPLEX(1,0),IMSIN(COMPLEX(B2,C2))) — cosecant as reciprocal of sine.

For complex arguments z=a+bi, i·z=-b+ai and -i·z=b-ai, explaining the signed pairs (-C2,B2) and (C2,-B2) in the Euler formulas. Check each nested function’s own official argument rules before modifying these expressions. Related functions are linked below.

Mathematical domain: Poles occur when sin(z)=0: real z=kπ, including z=0. The numerical result near a pole can be large and unstable. Its mathematical denominator is sin(z); zero input is mathematically undefined.

Numerical limits: Floating-point calculations can lose precision near zeros, poles or when hyperbolic sine/cosine grows rapidly. The mathematical values in the table are not guaranteed exact output formatting. The vendor’s one-paragraph IMCSC help entry does not document a specific overflow threshold, rounding mode, sign-of-zero policy or error-code mapping for all invalid inputs. Those require targeted app tests.

Do not label mathematically undefined as an observed spreadsheet error. csc(0) is mathematically undefined. A vendor help page that omits error behavior is not proof of a specific displayed error code. At zero and integer multiples of π on the real axis, cosecant has mathematical poles; do not invent native displayed codes.

The current ONLYOFFICE function catalogue and IMCSC help entry list the function in the unversioned current Docs help. This supports the narrow claim that the current documentation describes IMCSC. It does not establish the first shipping version, parity among self-hosted Docs, cloud Docs, Desktop Editors, mobile editions, or all historical versions.

ONLYOFFICE publishes a Docs changelog; its 10.0.0 release is dated October 10, 2026. That changelog is a release-history reference, not evidence that IMCSC debuted in 10.0. No release-specific function-introduction claim is made here. The current Docs manual explains that the Formula tab and Insert Function interface list available functions, so checking the actual installed product and build is the appropriate remaining compatibility test.

The supplied expressions use standard-looking spreadsheet function names, but another spreadsheet application’s behaviour is not ONLYOFFICE evidence. Opening an .xlsx, .ods or .xls file, calculated-value caching, localized separators, i/j suffix rendering and cross-editor numerical equality must each be tested directly before declaring compatibility. In particular, support for formula evaluation and preservation of stored results are different questions.

Native verification checklist still outstanding: date-stamped Desktop version, date-stamped Docs/web version, install/edition, locale and decimal separator, i/j rendering, invalid input and singularity behavior, tolerance at defined examples, exported .xlsx and .ods round-trip.

• IMSIN — official ONLYOFFICE function reference
• IMDIV — official ONLYOFFICE function reference
• IMPRODUCT — official ONLYOFFICE function reference
• COMPLEX — official ONLYOFFICE function reference
• IMREAL — official ONLYOFFICE function reference
• IMAGINARY — official ONLYOFFICE function reference

Calculation method: cmath (Python standard library) was checked against mpmath with 65-decimal precision for each defined table row; additional identity-based checks were applied where denominators are nonzero. These are independent mathematical calculations, not ONLYOFFICE executions. The official help documents purpose and syntax; it is not the source of the independently calculated six-decimal results.