IMCSC in Google Sheets: Validate Complex Cosecant with Reciprocal Controls

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

=IMCSC(number)

Google explicitly permits real numeric values, COMPLEX results and numeric x+yi strings. A numeric real value has imaginary part zero.

The current English reference says an error occurs for an invalid complex input or when cosecant is undefined. It does not name the native error code. This is more specific than a generic invalid-text warning, but it is not a complete numerical boundary table.

Cosecant is reciprocal sine, not inverse sine. The argument’s real coordinate is a radian angle in the mathematical model.

Enter the first two numeric columns in A2:B5. Use =COMPLEX(A2,B2) in C2 and =IMCSC(C2) in D2, filling down. Extract D’s real and imaginary coefficients with IMREAL and IMAGINARY into E:F.

Output columns are rounded independent expectations, not values to paste into A:B. The first row has real coefficient greater than one, while the second has smaller coefficients despite its larger imaginary input.

The reciprocal responds to the complete sine denominator. Its size is not a monotone function of either original coordinate alone.

Compare D2 with:

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

The independently expected quantities match for all four selected rows. IMSIN provides the denominator, whose first-row coefficients are approximately 0.703051146+0.184833620i.

Its squared magnitude is approximately 0.528444382. The reciprocal divides the conjugate coefficients by that shared number. This gives a negative imaginary coefficient even though the sine’s imaginary coefficient is positive.

Taking separate reciprocals of the two coordinates would calculate a different quantity. Check both coefficients after IMDIV rather than comparing only real parts.

A separately derived expression uses denominator COSH(2b)−COS(2a), with real numerator 2SIN(a)COSH(b) and imaginary numerator −2COS(a)SINH(b).

For the first row, use:

=2*SIN(0.75)*COSH(0.25)/(COSH(0.5)-COS(1.5))
=-2*COS(0.75)*SINH(0.25)/(COSH(0.5)-COS(1.5))

Expected coefficients are 1.330416541 and −0.349769298. This equation is independently derived mathematics, not a displayed equation in the Google reference.

For these moderate inputs, compare the result to D using both component differences and a declared tolerance such as 1E-9. An alternative formula is not a universal numerical-stability guarantee.

For each row, =IMPRODUCT(D2,IMSIN(C2)) has mathematical target one with imaginary coefficient zero. This catches a wrong imaginary sign that a magnitude-only comparison could miss.

The complete cosecant square also equals one plus the complete IMCOT square. For C2, either side has independently expected coefficients approximately 1.647669611−0.930677719i.

Use IMPRODUCT to calculate a complex square. Squaring only IMABS(D2) discards the phase and is not the same operation.

A real thirty-degree control is =IMCSC(RADIANS(30)); Google’s numeric-input rule permits it. Its mathematical target is two, with imaginary coefficient zero.

For purely imaginary 0.1i and 0.25i, the independent quantities are approximately −9.983352757i and −3.958635163i. They follow negative i divided by SINH(b).

The large first magnitude illustrates reciprocal amplification near a zero denominator. It does not establish a native precision threshold or justify treating every tiny nonzero input as exactly zero.

At exact mathematical zero, sine is zero and cosecant is undefined. That falls within Google’s documented undefined-function error condition, without assigning an error code. A formula involving PI() is a finite approximation and does not establish the same exact symbolic boundary.

The official numeric =IMCSC(0.213) example is approximately 4.73052448711521. Its 2−2i example has coefficients approximately 0.244687073586957−0.107954592221385i, corroborated independently.

Blank or logical coercion, mixed representations, underflow and extreme overflow are not fully specified by the dedicated reference. No native TYPE classification is inferred from a scalar-looking display.

• Google IMCSC 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.