Zoho Sheet COTH Function: Hyperbolic Cotangent with Worked Examples

• Introduced: The first supported release and original introduction date are not established by the checked sources.
• Current support: Listed in Zoho Sheet’s rolling web-service help, retrieved October 15, 2026. No numbered final functioning version or retirement date is established.
• Platform scope: English-language web documentation; historical clients, mobile and offline parity have not been verified.
• Evidence: Zoho’s COTH entry supports the syntax and vendor examples. The additional datasets below are independently calculated mathematical expectations. No formula was executed in native Zoho Sheet for this guide.

COTH(number)

Zoho documents one required numeric argument and no optional parameters. Supply a number directly, through a numeric expression, or from a cell. Unlike COT’s angle, this argument is a hyperbolic-function input; do not automatically convert it from degrees. A ratio of two lengths in the same unit is already dimensionless.

Mathematically, coth(x) = cosh(x)/sinh(x) = 1/tanh(x) for nonzero real x. COSH, SINH and TANH are the documented helpers used for the checks below. COTH is not ACOTH, which performs the inverse operation, and it is not COT, which is periodic trigonometry.

A single-argument call does not require commas or semicolons. The literal decimals shown use English punctuation; match the spreadsheet’s region when entering numbers.

Create CothControls, with Input in A1 and Output in B1. Enter 0.3, -0.9 and =1/2 in A2:A4. Enter =COTH(A2) in B2 and fill through row 4.

Zoho publishes these three input/result pairs; the trailing zero in the first result simply sets a consistent nine-decimal display. The values have also been independently calculated. They are not screenshots or observations from a Zoho workbook.

Create CothComparison, reserving A1:F8. Use headers Input, COTH, Reciprocal_TANH, COSH_over_SINH, Difference_1 and Difference_2 in A1:F1. Enter the values below in A2:A8, then fill these formulas from row 2 through row 8:

B2: =COTH(A2)
C2: =1/TANH(A2)
D2: =COSH(A2)/SINH(A2)
E2: =B2-C2
F2: =B2-D2

E and F are mathematically zero. Different evaluation paths can leave small last-digit residuals, so agreement here is a numerical check, not proof of identical algorithms. A hand-written exponential formula can introduce additional overflow or cancellation; it is unnecessary for these ordinary inputs.

For real nonzero x, coth is odd: changing the sign reverses the result’s sign. Its magnitude exceeds 1 mathematically, approaching 1 as the input’s magnitude grows. A finite-precision calculation can eventually display exactly 1 or -1 after rounding; that does not establish an exact mathematical limit was attained.

Duplicate CothComparison as ChangedInput and change only A6 from 0.5 to -0.5. B6/C6/D6 become approximately -2.163953414. Other rows remain as before. This control distinguishes a row-linked formula from a fixed literal.

Suppose a hypothetical model specifies output = scale*coth(length/reference_length). This is a demonstration of the supplied equation, not a claim that it describes a particular material or engineering system.

Create CothModel, with headings Length, Reference_length, Ratio, Factor, Output and Scale in A1:F1. Set A2:A4 to 1,2,4, B2:B4 to 2, and F2:F4 to 3. Fill these formulas through row 4:

C2: =A2/B2
D2: =COTH(C2)
E2: =F2*D2

Keep both length columns in the same unit. Duplicate the model as UnitConversion and multiply every A and B input by 100. Ratios, factors and outputs remain unchanged. Changing only one length column by 100 would alter the model instead of merely converting its units.

In a separate copy, double F2:F4 to 6. Each output doubles; for the ratio-1 row, the result is approximately 7.878211713. This gives two different checks: unit conversion preserves the ratio, while changing the scale changes output proportionally.

Zero is a mathematical singularity because sinh(0) is zero. The checked Zoho entry does not establish the exact runtime error token for COTH(0). Keep zero outside this finite-result model, and do not replace it with a fabricated zero output.

Near zero, the result is large: independently, COTH(0.1) is approximately 10.033311132, while COTH(0.01) is approximately 100.003333311. The output’s magnitude therefore does not imply a bad formula by itself. Examine whether the small ratio is intended and whether input measurement precision supports the result.

The official page’s common error glossary covers invalid types, names and references. It does not document every blank, text, array or extreme-input case. Exact coercion, maximum magnitude, native errors and behavior in historical/mobile clients remain unverified. Before wrapping the calculation in an error fallback, confirm both the main input and any denominator used to produce it.

COT takes a circular angle and has different poles. TANH gives the reciprocal relationship for nonzero inputs; ACOTH reverses the hyperbolic operation within its own real domain. COSH and SINH provide an explicit ratio check.

Matching function names in another spreadsheet do not certify input coercion, error displays, numeric limits, platform availability or export fidelity. Recheck the target application with the small control table before transferring a consequential model.