IMSIN in Google Sheets: Check Complex Sine with Real and Hyperbolic Helpers

• 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 IMSIN reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMSIN(number)

The argument is required. Google demonstrates both COMPLEX(4,1) and the real number 3.5. Invalid complex input is documented to produce an error, but this reference does not identify its exact code.

One worked-table formula places the equals sign after IMSIN. Enter the normal worksheet form =IMSIN("3+2i"); the misplaced equals sign is a presentation defect, not a distinct formula syntax.

The independent component model for x+yi is SIN(x)×COSH(y) for the real coefficient and COS(x)×SINH(y) for the imaginary coefficient. This mathematical equation checks the outputs below; the dedicated Google page does not print it.

Enter the first column as Plain text in A2:A4. Put =IMSIN(A2) in B2 and fill down. Use IMREAL and IMAGINARY to extract coefficients in C and D.

For a separate numeric control, =IMSIN(0.5) has independently expected real result approximately 0.479425539, matching ordinary SIN(0.5).

Exchanging the real and imaginary input coordinates does not merely exchange the output coordinates. The first two rows use ordinary and hyperbolic functions in different positions, so both output values change substantially.

For 2+0.5i, calculate:

=SIN(2)*COSH(0.5)
=COS(2)*SINH(0.5)

Expected coefficients are approximately 1.025347389 and −0.216852163. The imaginary sign is negative because COS(2) is negative while SINH(0.5) is positive.

A cell-driven version replaces two with IMREAL(A2) and 0.5 with IMAGINARY(A2). COMPLEX combines the numeric helper outputs into a complete quantity.

SINH and COSH are used on numeric coefficients. Applying ordinary sine separately to both input coordinates would be a different calculation.

Google gives a constructed-complex control, a real numeric control and a quoted complex control. Their independent expected coefficients are:

• =IMSIN(COMPLEX(4,1)): approximately −1.167807275 and −0.768162763.
• =IMSIN(3.5): real result approximately −0.350783228.
• =IMSIN("3+2i"): approximately 0.530921086 and −3.590564590.

The last line uses the corrected worksheet equals-sign placement. These controls support the documented forms without claiming acceptance of every arbitrary string, Boolean or empty reference.

Use the full formula results when making comparisons. The nine-decimal displays here intentionally shorten the independently calculated values.

For a real 45-degree angle, enter =IMSIN(RADIANS(45)). The expected real result is SQRT(2)/2, approximately 0.707106781.

For the pure-imaginary value 0+0.5i, the independent sine model has real coefficient zero and imaginary coefficient SINH(0.5), approximately 0.521095305.

These two inputs answer different questions. A degree format applied to a numeric cell does not change a radian function’s input unit, and a purely imaginary value is not an ordinary real angle with unusual formatting.

For z=2+0.5i, compute s=IMSIN(z) and c=IMCOS(z). The mathematical sum of their complete complex squares is one.

Use:

=IMSUM(IMPRODUCT(IMSIN(A2),IMSIN(A2)),IMPRODUCT(IMCOS(A2),IMCOS(A2)))

Expected target coefficients are one and zero. IMPRODUCT squares each whole complex value, including its cross terms.

Squaring the two magnitudes and adding instead gives COSH(1), approximately 1.543080635, for this input. It is a different expression; the imaginary coordinate cannot be discarded before verifying the full identity.

The real sine interval from −1 to one applies to real arguments. A valid complex sine may have a real coefficient or magnitude outside that interval, as the first two table rows show. Do not clamp them as a cleanup operation.

Google’s invalid-complex-input error statement is retained without inventing a code. Detailed blank/Boolean coercion, malformed-string precedence and extreme numeric behavior remain unverified.

For these modest independent controls, an absolute component tolerance of 0.000000001 is a useful declared check. It is not a service-wide precision guarantee, and no native Sheets execution was performed.

• Google IMSIN 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.