IMTAN Function (Gnumeric)
Learn the IMTAN function in Gnumeric: Returns the tangent of a complex number.
Verification status: Official documentation referenced; example outcomes independently reasoned or arithmetically checked where stated. Not executed in Gnumeric. How formula examples are checked
Evidence: Examples are independent mathematical calculations. Official documentation and the Gnumeric 1.12.62 source were inspected. No native Gnumeric workbook execution or application screenshots were produced.
What IMTAN does
Calculate the tangent of a complex number. Conceptually it is complex sine divided by complex cosine; both real and imaginary components can contribute to the quotient.
Gnumeric’s official Complex manual documents this exact function name and signature; the official function catalogue separately lists it as Complete with Exhaustive upstream testing. “Complete” describes the catalogue’s implementation classification; “Exhaustive” describes the listed upstream tests, not a claim about the example worksheet below or every available platform/version. Official documentation, checked 2026-10-13.
Version-specific error distinction: The online 1.12 manual describes malformed complex inputs as
#VALUE!. The official 1.12.62 source instead returns#NUM!when parsing a malformed complex string; other unsupported value types can still return#VALUE!. Any#VALUE!expectations below are the manual’s older documented rule, not verified 1.12.62 output. No native error test was performed.
Exact syntax and arguments
=IMTAN(z)
| Position | Argument | Required? | Type and meaning | Default |
|---|---|---|---|---|
| 1 | z |
Yes | One complex number, for example "1+2i", or a cell containing a complex-number representation |
None; there is no optional argument |
There are no additional arguments. z stands for a single complex value, not two separate real-number arguments. To construct a value from distinct coefficients use the separately documented COMPLEX function; in Gnumeric the related function’s arguments are COMPLEX(x,y,i) where the optional i/j suffix defaults to i. The simple one-argument examples here work independently of list-separator conventions, but COMPLEX’s comma notation reflects the English manual; check the installed locale before pasting multi-argument expressions. Official source, checked 2026-10-13.
Radian convention. Interpret angular inputs in the complex trigonometric formulas using radians. The real and imaginary components of the input are ordinary numeric coefficients; appending i marks the imaginary coefficient, not an angle unit. If starting with a real angle in degrees, convert it to radians first before constructing z; do not append i to a degree number and expect automatic conversion.
The mathematics behind the formula
Definition: tan(z) = sin(z) / cos(z) wherever cos(z) ≠ 0. One equivalent real/imaginary expression is tan(x+iy) = [sin(2x) + i sinh(2y)] / [cos(2x) + cosh(2y)]. The expression is useful for understanding growth and near-pole behavior.
The mathematical formula explains the independent expected outputs below. It is not a separate assertion that an untested Gnumeric build has identical last-digit floating-point behavior in every environment.
Reproducible beginner worksheet
Create an empty Gnumeric sheet with column headings: A = complex input, B = IMTAN result, C = real component, D = imaginary coefficient. Enter the text values in A2:A6 (if automatic cell conversion interferes, explicitly format A cells as text), then enter the formulas in B2:D2 and fill them down. Do not type the illustrative expectation into the formula-result cells.
| Cell | Enter this value or formula | Independently calculated expectation |
|---|---|---|
| A2 | 1+2i (text) |
Complex argument |
| B2 | =IMTAN(A2) |
Complex result; text formatting may vary |
| C2 | =IMREAL(B2) |
Real component ≈ 0.033812826 |
| D2 | =IMAGINARY(B2) |
Imaginary coefficient ≈ 1.014793616 |
| A3 | 0+0i (text) |
Complex argument |
| B3 | =IMTAN(A3) |
Complex result; text formatting may vary |
| C3 | =IMREAL(B3) |
Real component ≈ 0.000000000 |
| D3 | =IMAGINARY(B3) |
Imaginary coefficient ≈ 0.000000000 |
| A4 | 2-1i (text) |
Complex argument |
| B4 | =IMTAN(A4) |
Complex result; text formatting may vary |
| C4 | =IMREAL(B4) |
Real component ≈ -0.243458201 |
| D4 | =IMAGINARY(B4) |
Imaginary coefficient ≈ -1.166736257 |
| A5 | 0+1i (text) |
Complex argument |
| B5 | =IMTAN(A5) |
Complex result; text formatting may vary |
| C5 | =IMREAL(B5) |
Real component ≈ 0.000000000 |
| D5 | =IMAGINARY(B5) |
Imaginary coefficient ≈ 0.761594156 |
| A6 | 3+4i (text) |
Complex argument |
| B6 | =IMTAN(A6) |
Complex result; text formatting may vary |
| C6 | =IMREAL(B6) |
Real component ≈ -0.000187346 |
| D6 | =IMAGINARY(B6) |
Imaginary coefficient ≈ 0.999355987 |
These values are rounded to nine decimal places. A complex-result cell may display additional digits, omit a zero coefficient, or represent imaginary units with another supported suffix; the table tests numeric coefficients, not exact string serialization. The official manual documents IMREAL and IMAGINARY for extracting those parts. If the input entered as 1+2i is not treated as a valid complex text representation on a particular system, explicitly enter it as a string or generate it with =COMPLEX(1,2); do not silently assume every locale parses input the same way. Documentation checked 2026-10-13.
A direct formula to try
=IMTAN("1+2i")
Use =IMREAL(IMTAN("1+2i")) and =IMAGINARY(IMTAN("1+2i")) to compare the two independently expected coefficients from the table above. Both the compound formulas and the output-format details are review examples pending native confirmation, rather than proof that they were executed in Gnumeric.
Practical scenario
A wave-model input of 1+2i has independent expected tangent 0.033812826 + 1.014793616i. Its imaginary component is the dominant one. The result is not a tangent of a pair of independent real numbers.
Advanced use and verification identities
For a method comparison, evaluate these two formula paths in separate cells with A2 = 1+2i:
=IMTAN(A2)
=IMDIV(IMSIN(A2),IMCOS(A2))
The mathematical results agree whenever the denominator is nonzero, but floating-point implementations may produce slightly different last digits. Compare real and imaginary coefficients numerically rather than matching complex-value text strings. A useful theoretical limit is tan(iy) = i tanh(y): for "0+1i", the independently computed imaginary coefficient is approximately 0.761594156. The companion IMTANH is available as a related calculation.
Errors and troubleshooting
| Symptom or issue | What is supported by evidence, and what to check |
|---|---|
#VALUE! for invalid z |
The official Complex manual expressly documents this response when its complex-number argument is invalid. Check for malformed expressions such as an extra i, unsupported embedded spaces, wrong quotation marks, or mistakenly split real and imaginary inputs. Exact parsing of every malformed text variant was not tested. |
| A cell appears to return unexpected text | First inspect whether A2 is a text value and whether the result is a complex representation. Extract real and imaginary components with IMREAL/IMAGINARY rather than treating the complex result as an ordinary scalar or comparing formatted strings. |
| Real-vs-complex confusion | Assuming complex tangent has all the same numerical limitations as the ordinary real-domain TAN formula. Complex cosine can approach zero and make the tangent very sensitive to small perturbations. Near a pole, a seemingly small input adjustment can generate a huge result. |
| Extreme input magnitude or instability | Mathematically, tan(z) has poles at real z = π/2 + kπ for integer k (imaginary part zero). Finite floating-point approximations near these poles may yield extremely large finite values instead of a clean division-by-zero error. We did not establish which precise Gnumeric runtime error is returned for such inputs; do not promise #DIV/0!. |
| Wrong values after copying between sheets/applications | Verify the active locale, the interpretation of i/j suffixes, function availability in the target product and the source application’s actual version. Excel and LibreOffice behavior is not independently established Gnumeric runtime evidence. |
| A blank or numeric-only input behaves surprisingly | Official documentation establishes the requirement for a valid complex number, but it does not specify all blank-cell coercion and text-parsing cases. Test these directly in the installed Gnumeric version instead of inferring output from Excel or a mathematical library. |
Numerical tolerance tip: even for mathematically exact identities such as a sum being 1+0i, roundoff can leave a very small residual. Comparing text representations for equality is not a sound numerical test. Set a problem-appropriate absolute/relative tolerance for each numeric coefficient.
Compatibility, software version, and platform evidence
The official Gnumeric Complex manual explicitly describes IMTAN as Microsoft Excel compatible. Near-pole rounding, internal precision, complex suffixes and all error cases remain version-dependent and were not validated by a native cross-software run.
The official release material identifies Gnumeric 1.12.62 in September 2026, and the GNOME source archive lists the corresponding 1.12.62 tarball. This review uses the online official manual and catalogue as consulted 2026-10-13, alongside inspection of checksum-verified official 1.12.62 source; no source build was executed. Consequently, the evidence supports documented function availability in the contemporary Gnumeric project, not a claim that every Linux distribution, platform port, or historical build produces identical results. Official release notes · GNOME release archive.
Related Gnumeric function guides
Sources, checked on 2026-10-13
- Official Gnumeric Complex function manual — exact
IMTAN(z)signature, argument meaning, invalid-input#VALUE!note, specific compatibility wording where provided, and nearby related complex functions. - Gnumeric’s public function catalogue — contemporary listing and labels “Complete” and “Exhaustive”; these are classification/testing labels, not a runtime experiment by this research task.
- Official Gnumeric 1.12.62 release announcement — project version and general Excel-compatibility improvements; not proof of this function’s exact release implementation.
- GNOME source-release directory — available 1.12.62 source archive; official 1.12.62 source inspected after checksum verification; not compiled or executed.
Evidence limits: Official documentation and release-source inspection; independent mathematics. No native Gnumeric execution, application screenshots, locale tests or cross-application workbook round-trips.