Zoho Sheet IMSIN: Complex Sine with Signed Examples

• 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 IMSIN reference. Documentation and independent complex arithmetic; no native Zoho Sheet test.

=IMSIN(complex_number)

The single required argument is complex-number text. Use a quoted literal inside a formula, or a reference to a cell containing that text. There is no optional degree switch, return mode or separate suffix argument.

For z=a+bi, the independent coefficient identity is:

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

The real trigonometric terms take radians. The hyperbolic terms are what make this different from calculating ordinary sine separately for each component. Zoho’s published examples agree with this identity, including a negative imaginary coefficient and a pure imaginary input.

Create a new sheet named Sine. Enter Complex input, a, b, Complex sine, Result real, Result imaginary, Model real and Model imaginary in A1:H1. Fill A2:C5 with the following values. The entries in A are cell text without enclosing quote characters.

Put =IMSIN(A2) in D2, =IMREAL(D2) in E2 and =IMAGINARY(D2) in F2. Fill D2:F2 through row 5. The first two input forms reproduce Zoho’s examples; the other rows are independent mathematical checks.

Keep full results in the cells and use formatting to display eight decimals. E:F isolate the coefficients so that a change in the displayed complex string’s precision does not become a false disagreement.

Enter =SIN(B2)*COSH(C2) in G2 and =COS(B2)*SINH(C2) in H2. Fill G2:H2 through row 5. Compare G with E and H with F.

For 1−4i, cosh(−4) is positive while sinh(−4) is negative. Since sine and cosine of 1 radian are positive, the resulting real coefficient is positive and the imaginary coefficient negative. This explains the signs without trying to split the returned text at every plus or minus character.

The numeric a and b columns are a separate representation of the original input. If you edit A, update B:C consistently before using the helper formulas for validation.

Put =IMSIN("5j") in J2. Zoho reports approximately 74.2032105777888j. In J3 enter =SINH(5); its independent expected numeric value is 74.2032105777888, the coefficient of j in the complex result.

Sine of a real angle is bounded by −1 and 1, but sine of a complex value need not be. Here the result is purely imaginary and its magnitude exceeds 74. Do not replace IMSIN with SIN(IMREAL(…)); that would discard the information responsible for this result.

The zero real component may be omitted from the displayed representation. Compare its mathematical components rather than requiring an explicit 0+ prefix. The source does not fully specify every internal datatype or output-format edge case.

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

In F2 enter =C2+C3 and in F3 enter =D2+D3. Both mathematical targets are 0. For these modest values, a tolerance of 0.000000001 is a reasonable comparison scale rather than demanding exact text equality.

This tests the odd identity sin(−z)=−sin(z). Both input coordinates must change sign. Changing 1+i to 1−i tests a different relationship, so do not confuse a sign change in one component with negating the complete value.

Check function spelling, quotation marks and source cell references before changing the mathematics. The help lists general errors such as invalid types and names, but does not define a result for every blank, malformed string or overflow input. In its related-function list the page misspells COMPLEX as COMLPEX; the documented constructor is COMPLEX, not that typo.

Use IMCOS for complex cosine. The real-valued helpers SIN, COS, COSH and SINH support the independent decomposition. Different products can use different parser, error and formatting rules even where function names match.

Official sources: IMSIN, IMREAL, IMAGINARY, COSH and SINH.