IMCSC in Collabora Office: Audit Complex Cosecant as a Complete Reciprocal

• Introduced: the original Collabora release and launch date are not established by the reviewed sources.
• Availability: documented in Collabora Office 26.04 desktop help.
• Platform scope: online and mobile deployment support is not inferred from the desktop documentation.
• Checked: October 14, 2026. Function-specific documentation and independent controls; no native Collabora execution.

=IMCSC(Complex_number)

The documented input forms are a+bi and a+bj text. A purely real argument may also be a numeric value. The result is explicitly described as a complex string, omitting any zero part from display.

For example, the reference’s =IMCSC(2) returns approximately 1.09975017029462 as text. A later numeric scalar calculation should extract the real coefficient explicitly rather than assume that the displayed decimal changed the return type.

Cosecant is a reciprocal function, not an inverse sine that returns an angle.

Format A2:A5 as Text and enter 0.5+0.75j, 2+j, 0.5-0.75j and 2j. Calculate B with =IMCSC(A2) filled down. Use IMREAL and IMAGINARY on B to populate C and D.

Values are independent expectations rounded to nine decimals. The first and third rows are conjugates; the last has a purely imaginary reciprocal with a negative coefficient.

For A2, compare IMCSC with:

=IMDIV(COMPLEX(1;0;"j");IMSIN(A2))

The expected quantity matches the primary table. The numerator is explicitly constructed as a complex one, avoiding an assumption about implicit numeric coercion by IMDIV.

If sine has real coefficient r and imaginary coefficient s, its reciprocal has coefficients r/(r²+s²) and −s/(r²+s²). For the first input, sine is approximately 0.620704231+0.721650824j, and its squared magnitude is approximately 0.906053655.

Dividing those coefficients by the shared denominator, with the imaginary sign reversed, gives the cosecant components. It is not correct to take 1/r and 1/s separately.

Collabora’s equation has denominator cosh(2b)−cos(2a). Its real numerator is 2sin(a)cosh(b), and its imaginary numerator is −2cos(a)sinh(b).

For a=0.5 and b=0.75, check:

=2*SIN(0.5)*COSH(0.75)/(COSH(1.5)-COS(1))
=-2*COS(0.5)*SINH(0.75)/(COSH(1.5)-COS(1))

Expected coefficients are 0.685063437 and −0.796476920, rounded. This equation is consistent with the reciprocal’s squared-magnitude denominator, expressed using double-angle identities.

For each primary row, use =IMPRODUCT(IMCSC(A2);IMSIN(A2)) and fill down. The independent mathematical target is one, with zero imaginary coefficient.

IMPRODUCT checks the complete reciprocal relationship. For the first row, the cosecant magnitude is approximately 1.050565275, equal to one divided by the sine magnitude. A magnitude-only check is useful, but would not catch every wrong imaginary sign; check components too.

IMSIN supplies the denominator, and IMABS supplies a real magnitude for this secondary control.

For a real thirty-degree angle, =IMCSC(COMPLEX(RADIANS(30);0;"j")) has mathematical real coefficient two, with zero imaginary coefficient.

For imaginary inputs 0.1j, j and 2j, expected quantities are approximately −9.983352757j, −0.850918128j and −0.275720565j. On this axis, the model is negative j divided by sinh(b), not ordinary CSC(b).

The small 0.1j input illustrates reciprocal amplification near a zero of sine. It does not define a native threshold or certify arbitrary accuracy as the input approaches zero.

Cosecant has mathematical poles where sine is zero, including real integer multiples of pi. Exact native zero-input errors and results at a floating approximation to PI() are not established by the dedicated page.

Its 4−3i example gives coefficients approximately −0.0754898329158637 and −0.0648774713706355. The captured equation and an independent reciprocal calculation agree with that control.

The reference’s OpenDocument 1.2 statement does not date Collabora’s first release. Mixed suffixes, malformed text, blank coercion and overflow details remain unverified. Use explicit component tolerances for the moderate examples rather than claiming exact native string equality.

• Collabora Office 26.04: IMCSC.
• Editorial testing standards.

Complete dedicated reference and its equation image reviewed. Examples are independently calculated controls or explicit documented statements, not native application captures. Formulas use English names and semicolon separators.