IMCOS in Excel: Calculate Complex Cosines with Correct Imaginary Signs

• Introduced: the first introduction version is not established by the reviewed official history.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2019 for Mac, Excel 2016.
• Checked: October 14, 2026; official IMCOS reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMCOS(inumber)

The complex input is required. Microsoft describes i or j text and recommends COMPLEX for numeric coefficients.

The imaginary term has a minus sign. Omitting it produces a conjugated value for many ordinary inputs, which can look plausible while breaking subsequent complex calculations.

Microsoft explicitly documents #VALUE! when inumber is a logical value. This named rule does not establish the entire blank, numeric-text or malformed-input matrix. The examples use valid constructed values or complex text.

Enter the first column as text in A2:A5. Put =IMCOS(A2) in B2 and fill down. Extract the result’s coefficients into C and D.

Microsoft’s explicit 1+i example supports the second row. Reversing the imaginary input coefficient produces the third row’s conjugate result.

For the pure-imaginary fourth row, COS(0)=1 and SIN(0)=0, so the real result is COSH(1). It exceeds one, which is valid for a complex argument.

Use =IMREAL(B2) and =IMAGINARY(B2) for numeric inspection. Do not rely on matching the exact number of decimal digits in a displayed complex string.

For z=1+i, enter:

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

The expected coefficients are approximately 0.833730025 and −0.988897706. COSH and SINH operate on numeric helper arguments in this equation.

A cell-driven version uses IMREAL(A2) for the ordinary trigonometric argument and IMAGINARY(A2) for the hyperbolic argument. Ordinary sine and cosine take radians.

The source equation supports this reconstruction for valid complex inputs, but does not certify numerical equality of all hand-written implementations at extreme scales.

For a real 60-degree angle, =IMCOS(COMPLEX(RADIANS(60),0)) has mathematical real coefficient 0.5 and imaginary coefficient zero.

For the imaginary-axis input i, the expected real coefficient is COSH(1), approximately 1.543080635. These are different directions in the complex plane, even though each fixture contains the numeric coefficient one somewhere.

The real-only cosine range from −1 to one should not be imposed on arbitrary complex results. In particular, clipping the i result to one would destroy the documented equation’s value.

A degree label in cell formatting does not change the input unit. Convert degrees explicitly before placing a real angle into COMPLEX.

Negating both coordinates of 1+i gives −1−i. In the independent model, their complex cosines are equal: both have coefficients approximately 0.833730025 and −0.988897706.

This differs from conjugation, which changes only the imaginary sign. The cosine of 1−i is the conjugate result, with positive imaginary coefficient approximately 0.988897706.

These controls help distinguish three operations that are easy to confuse in a formula: negating the input, conjugating the input and negating the output.

For z=1+i, square IMCOS(z), square IMSIN(z), then add with IMSUM. The mathematical target has real coefficient one and imaginary coefficient zero.

The cosine square has coefficients approximately −0.282812918 and −1.648947418. The sine square has coefficients approximately 1.282812918 and 1.648947418. Their imaginary parts cancel.

This identity involves complex multiplication through IMPRODUCT. Squaring only the real coefficients would miss the other components and would not verify the same statement.

Similarly, summing the squares of the two magnitudes gives COSH(2), approximately 3.762195691, rather than one. State exactly which operation a checking column performs.

A logical TRUE or FALSE supplied as inumber follows Microsoft’s documented #VALUE! rule. An actual error in a source cell is a different case, and error precedence is not established by that single rule.

For these moderate fixtures, a component tolerance of 0.000000001 is a useful independent comparison. Large hyperbolic factors can cause overflow or loss of accuracy in identities with cancellation; no native threshold is certified here.

The page does not provide a complete malformed-text, whitespace or degenerate storage-type specification. All calculations shown are independently modeled, with no native Excel execution.

• Microsoft IMCOS 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.