Zoho Sheet IMCOS: Complex Cosine with Component Checks

• Introduced: The original introduction release and date are not established by the checked sources.
• Availability: Currently documented in Zoho Sheet’s rolling web-service help; no numbered release or retirement endpoint is established.
• Scope: English-language web-service documentation. Mobile and offline-client parity is outside the verified scope.
• Checked: October 14, 2026, against the official IMCOS reference. Documentation and independent complex arithmetic; no native Zoho Sheet test.

=IMCOS(complex_number)

The required argument is text representing a complex number, supplied as a quoted literal or a reference to a text cell. There are no optional degree, rounding or suffix arguments. Zoho’s examples use both i and j notation.

The official entry gives the identity:

cos(a+bi) = cos(a)*cosh(b) - i*sin(a)*sinh(b)

The real trigonometric terms use radians. The imaginary coefficient b enters the hyperbolic functions; it is not a second ordinary angle to feed into COS. Treat a and b as the two coordinates of the complex argument, not independent results to concatenate after applying ordinary cosine twice.

Create a new sheet named Cosine. In A1:H1 enter Complex input, a, b, Complex cosine, Result real, Result imaginary, Model real and Model imaginary. Enter the following text in A and numbers in B:C. Do not include quote characters around cell text.

Enter =IMCOS(A2) in D2, =IMREAL(D2) in E2 and =IMAGINARY(D2) in F2. Fill D2:F2 through row 5. The first two rows use Zoho’s documented inputs; the last two are independently calculated extensions of the same identity.

Column D holds the complex representation. Compare its extracted numeric coefficients in E:F rather than requiring an exact number of printed decimals or a particular text spelling for every result.

On Cosine, enter =COS(B2)*COSH(C2) in G2 and =-SIN(B2)*SINH(C2) in H2. Fill G2:H2 through row 5. G should agree with E and H with F to the displayed precision.

For the first row, cosine of 2 is negative and hyperbolic cosine of 3 is positive, making the real coefficient negative. Sine of 2 and hyperbolic sine of 3 are positive, so the leading minus sign makes the imaginary coefficient negative too. Losing that minus sign would produce a different complex value.

Keep B:C numeric and retain A as the source representation. If a source value changes, update its coordinate columns too before using the helper formulas as a check.

In J2 enter =IMCOS("5j"). Zoho’s published result is approximately 74.2099485247879. In J3 enter =COSH(5); its independent expected value is the same.

The input has a=0, so the identity reduces to cosh(5), with zero imaginary coefficient. The familiar range from −1 to 1 applies to cosine of a real angle, not to all complex inputs. A large complex-cosine result is therefore not automatically a calculation error.

The official example shows the pure-real output without a suffix. It does not establish the internal cell datatype or every zero-term formatting rule; this comparison uses the reported value rather than treating its spelling as a parser specification.

Create a separate sheet named Symmetry. Put Complex input, Cosine, Real coefficient and Imaginary coefficient in A1:D1. Enter 1+i in A2 and -1-i in A3. Enter =IMCOS(A2) in B2, =IMREAL(B2) in C2 and =IMAGINARY(B2) in D2. Fill B2:D2 through row 3.

Both rows have expected coefficients 0.833730025131149 and -0.988897705762865. Put =C3-C2 in F2 and =D3-D2 in F3; both mathematical targets are 0. Compare with a small tolerance such as 0.000000001 for these values rather than requiring exact text equality.

Cosine is even: negating the whole complex argument leaves its cosine unchanged. Negating just one coordinate is a different transformation. Preserve both minus signs in A3 so the example actually tests cos(−z), not a different input.

Zoho lists invalid types, invalid names and broken references among its possible errors, but does not enumerate a complete complex-string grammar. Check quoted literals, lowercase suffixes and units accidentally appended to inputs before diagnosing the formula. Blank strings, overflow boundaries and unusual numeric formats are not established here.

The related real functions COS, SIN, COSH and SINH explain the coefficient check. IMSIN calculates complex sine; COMPLEX builds a representation from numeric coordinates.

Official sources: IMCOS, IMREAL, IMAGINARY, COSH and SINH. After importing a workbook, compare values and input coordinates; shared names alone do not certify cross-product parser behavior.