IMCSC in Excel: Calculate Complex Cosecant and Check Reciprocal Signs

• Introduced: the Microsoft function catalogue marks this identity as an Excel 2013 addition. An exact first build or preview date is not established.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2019 for Mac, Excel 2016.
• Checked: October 14, 2026; official IMCSC reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMCSC(inumber)

Inumber is required and uses x+yi or x+yj notation. Microsoft recommends constructing the argument from numeric coordinates with COMPLEX. This also avoids hand-built strings with missing signs or an incorrectly placed imaginary unit.

Cosecant is a reciprocal, not an inverse sine. It does not recover an angle from a sine value. The real coordinate of the complex argument is measured in radians in the mathematical model.

The current dedicated page lists no Excel 2024 entry and no web client. The availability list above is reproduced as documented; this omission alone is not a claim that those clients reject the function.

Enter only the numeric input columns in A2:B5. Put =COMPLEX(A2,B2) in C2 and =IMCSC(C2) in D2; fill down. Extract with =IMREAL(D2) and =IMAGINARY(D2) in E and F.

The output columns are rounded independent expectations, not values to paste into A:B. Preserve D’s calculated complex result when evaluating the following checks.

The second input has a larger imaginary coordinate but a smaller output magnitude. Reciprocal size depends on the complete sine denominator, not on either original coordinate alone.

Use this alternative in G2 and fill down:

=IMDIV(COMPLEX(1,0),IMSIN(C2))

It should match the table’s complete quantities in the independent model. IMSIN supplies the denominator and IMDIV performs complex division.

For C2, the squared magnitude of sine is approximately 0.979613736. If that sine is r+si, its reciprocal has real coefficient r/(r²+s²) and imaginary coefficient −s/(r²+s²).

The imaginary sign changes through conjugation. Taking 1/r and 1/s separately is not the reciprocal of r+si and can fail even when both coordinates are nonzero.

An independently derived double-angle expression for cosecant uses denominator COSH(2b)−COS(2a). Its numerators are 2SIN(a)COSH(b) and −2COS(a)SINH(b).

For a=1 and b=0.5, use:

=2*SIN(1)*COSH(0.5)/(COSH(1)-COS(2))
=-2*COS(1)*SINH(0.5)/(COSH(1)-COS(2))

These return the mathematical targets 0.968610889 and −0.287408174, rounded. The expanded equation is a separate algebraic control; it is not presented as an equation printed on Microsoft’s dedicated page.

Compare both components within a stated tolerance, such as 1E-9 for this fixture. A test of only the magnitude would not detect an accidentally reversed imaginary sign.

For each row, calculate =IMPRODUCT(D2,IMSIN(C2)). The independent target is one, with zero imaginary coefficient.

For the first row, =IMABS(D2) is approximately 1.010351678. The independently equivalent check =1/IMABS(IMSIN(C2)) has the same target.

IMABS produces a real magnitude. Use that as a secondary check after the two component checks, not as a replacement for them.

The complete square relation also links cosecant to IMCOT: IMCSC(C2) squared equals one plus IMCOT(C2) squared. Both sides have expected coefficients approximately 0.855603596 and −0.556773374 for the first row.

For a real thirty-degree angle, use =IMCSC(COMPLEX(RADIANS(30),0)). Its mathematical target is real two with zero imaginary coefficient. Convert the angle once; do not pass thirty as though it already meant radians.

Imaginary inputs 0.1i, i and 2i instead have expected quantities near −9.983352757i, −0.850918128i and −0.275720565i. On that axis, cosecant is negative i divided by SINH(b).

The first value demonstrates amplification near a zero denominator. It does not establish an application’s error threshold or guarantee precision for arbitrarily tiny arguments.

Microsoft specifies #NUM! for unsupported complex-text format and #VALUE! for logical arguments. Plain-number coercion, blank treatment and extreme numerical limits are not fully documented by this page.

The mathematical poles occur where sine is zero, including real integer multiples of pi. Native errors at zero or at floating-point PI() formulas remain unverified here.

Microsoft’s =IMCSC("4+3i") example has coefficients approximately −0.0754898329158637 and 0.0648774713706355. An independent reciprocal calculation agrees, including the positive imaginary sign.

• Microsoft IMCSC 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.