IMCOSH in Google Sheets: Audit Complex Hyperbolic Cosines and Symmetry

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

=IMCOSH(number)

The number argument is required. Invalid complex input is documented to produce an error, but the dedicated reference does not identify an exact code or full coercion matrix.

The independent mathematical model for x+yi has real coefficient COSH(x)×COS(y) and imaginary coefficient SINH(x)×SIN(y). The imaginary sign is positive in this equation.

Google’s dedicated page supplies worked results rather than printing that component equation. The models below independently corroborate those results and show how to inspect the two coordinates separately.

Enter the first column as Plain text in A2:A4. Put =IMCOSH(A2) in B2 and fill down. Extract real and imaginary coefficients into C and D.

These are independent nine-decimal displays, not native captured strings. Keep the unrounded B values when checking later identities.

For a separate supported numeric input, =IMCOSH(0.5) has expected real result approximately 1.127625965, equal to ordinary COSH(0.5).

The second mixed input can have a negative real output because COS(2) is negative. The familiar positive behavior of real COSH inputs does not carry over unchanged to arbitrary complex values.

For the first row, calculate:

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

Expected coefficients are approximately 3.301637333 and 1.738809504. COMPLEX combines these helpers into a complete value.

The same model uses IMREAL(A2) as the hyperbolic argument and IMAGINARY(A2) as the ordinary trigonometric argument. Reversing those roles computes a different quantity.

The corresponding complex ordinary-cosine equation has a negative imaginary term. IMCOS and IMCOSH should therefore not be substituted for each other merely because their names are similar.

Hyperbolic cosine is half the sum of the exponentials of z and −z. For z=2+0.5i, compare:

=IMDIV(IMSUM(IMEXP("2+0.5i"),IMEXP("-2-0.5i")),COMPLEX(2,0))

Its independent expected coefficients equal the first table row. The negative exponent reverses both input coordinates, while the divisor is explicitly constructed as a real complex value.

IMEXP and IMSUM let both intermediate quantities remain visible. This structure is an independently checked definition, not evidence that every manually expanded formula shares the native function’s overflow behavior.

Keep the input scale modest when reproducing it. Large positive or negative real coordinates can produce huge intermediate exponentials even when a later expression combines them.

Negating both coordinates of 2+0.5i gives −2−0.5i. Hyperbolic cosine is even in the independent model, so its complete output is unchanged.

Conjugating changes only the imaginary sign. The input 2−0.5i therefore produces the conjugate output shown in the third table row, with a negative imaginary coefficient.

Those are different relationships. A formula that negates the whole output instead of applying the intended input transformation would fail these controls.

For the real input 0.5, the expected real result is COSH(0.5), approximately 1.127625965.

For input 0+0.5i, the independent output has real coefficient COS(0.5), approximately 0.877582562, and imaginary coefficient zero.

Equal input magnitudes do not imply equal hyperbolic cosines. Preserve direction when deciding whether a real-only reduction is appropriate for a particular record.

For z=2+0.5i, compute c=IMCOSH(z) and s=IMSINH(z). The mathematical difference c squared minus s squared is one.

Use IMPRODUCT to square each full complex result, then IMSUB to subtract. Check expected real coefficient one and imaginary coefficient zero, using a declared tolerance for approximate arithmetic.

Subtracting their squared magnitudes instead gives COS(1), approximately 0.540302306, for this input. That is a different valid identity. Magnitudes discard phase before subtraction, so they cannot replace the complete complex products.

Google documents IMCOSH of COMPLEX(4,1) with coefficients approximately 14.754701170 and 22.963673499. Its real 3.5 control gives approximately 16.572824671, and 3+2i gives approximately −4.189625691 and 9.109227894.

The independent component equation corroborates these controls. Exact invalid-input codes, Boolean/blank coercions, degenerate result TYPE and extreme precision remain unverified. No native Sheets execution was performed.

• Google IMCOSH 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.