IMTAN in Google Sheets: Verify Complex Tangents and Undefined-Input Limits

• Introduced: first introduction date is not established by the reviewed official history; no desktop release number is inferred.
• Availability: documented in the rolling Google Sheets service; these examples use the web worksheet workflow.
• Checked: October 14, 2026; official IMTAN reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMTAN(number)

The argument is required. Real numbers are interpreted as complex values with imaginary coefficient zero. This permits explicit numeric controls such as Google’s example using 0.213.

The source’s error note covers invalid complex values and inputs where tangent is undefined, but it does not name an exact displayed error code. A separate question is how a floating approximation to a mathematical pole is handled; that is not resolved by the generic note.

The independent mathematical model used below is complex sine divided by complex cosine where the denominator is nonzero.

Enter the first column as Plain text in A2:A4. Put =IMTAN(A2) in B2 and fill down. Extract real and imaginary coefficients into C and D.

The first and third rows are conjugate results. The second changes which coordinate is real, producing a different complete output.

For a separate real-number check, =IMTAN(0.5) has expected real result approximately 0.546302490, matching ordinary TAN(0.5).

Do not interpret a large imaginary tangent coefficient as an invalid result merely because it exceeds one. The first row is a valid independently modeled example.

For each input compare B with:

=IMDIV(IMSIN(A2),IMCOS(A2))

IMSIN and IMCOS produce complete complex numerator and denominator quantities. IMDIV combines both coefficients correctly.

For 2+0.5i, the sine has coefficients approximately 1.025347389 and −0.216852163. The cosine has coefficients approximately −0.469257978 and −0.473830620. Complex division yields the first table row.

Dividing only the real sine coefficient by the real cosine coefficient would not produce the complex tangent’s real coefficient. The denominator’s imaginary part affects both quotient coordinates.

For real x and y, a mathematical tangent model uses denominator COS(2x)+COSH(2y), real numerator SIN(2x), and imaginary numerator SINH(2y).

For x=2 and y=0.5, calculate:

=SIN(4)/(COS(4)+COSH(1))
=SINH(1)/(COS(4)+COSH(1))

Expected coefficients are approximately −0.850878121 and 1.321286584. This is an independent derivation; the dedicated Google IMTAN page does not print the identity.

The manual expression is useful for these moderate inputs. It does not certify equivalent native overflow or cancellation behavior for every possible coefficient size.

For input iy, the independent tangent model is i×TANH(y). With y=0.5, one and two, the expected imaginary coefficients are approximately 0.462117157, 0.761594156 and 0.964027580, with real coefficient zero.

Build those values using =COMPLEX(0,y), substituting an actual numeric cell for y. The real TANH function checks the numeric imaginary coefficient in this special case.

This special-axis relationship does not impose the same bound on every mixed complex input. Compare the first primary row, whose imaginary tangent coefficient exceeds one.

Negating both coordinates of 2+0.5i gives −2−0.5i. Tangent is odd in the independent model, so its expected coefficients are approximately 0.850878121 and −1.321286584.

Adding PI() to the real coordinate preserves the mathematical tangent. Compare =IMTAN(COMPLEX(2+PI(),0.5)) with the first row, using a stated component tolerance rather than exact formatted-string equality.

A full period in a formula is a numeric construction. Do not type an unevaluated expression such as PI() inside a quoted complex literal and assume it will be parsed as a worksheet calculation.

Google’s numeric control =IMTAN(0.213) gives approximately 0.216280749620508. Its quoted control =IMTAN("2-2i") has coefficients approximately −0.028392953 and −1.023835595.

For purely real arguments, mathematical poles occur at odd multiples of PI()/2. A floating-point PI()/2 expression is not exact symbolic input, so this guide does not promise a universal native pole error or finite result from it.

Invalid/undefined errors are documented at a general level. Exact codes, coercions, extreme precision and error precedence remain unverified. No native Sheets execution was performed.

• Google IMTAN reference.
• Editorial testing standards.

Examples use fictional data, English function names and comma separators. Expected outputs are independent mathematical calculations or explicitly documented rules, not native Sheets captures. Check the spreadsheet locale when reproducing punctuation-sensitive formulas.