COTH in Google Sheets: Calculate Hyperbolic Cotangent and Handle Zero

• Introduced: first service launch date is not established by the official history reviewed; no numbered desktop introduction is assigned.
• Availability: documented in the rolling Google Sheets service. Examples use ordinary web-spreadsheet numeric cells; other clients and imported files are not separately certified.
• Checked: October 14, 2026; official COTH reference. Documentation and independent calculations, without native Sheets execution.

=COTH(value)

Google describes a real numeric input. The mathematical ratio is COSH(value)/SINH(value), also 1/TANH(value) for nonzero ordinary real inputs. This is a hyperbolic operation, not circular cotangent of a degree angle.

No degree conversion is applied automatically to the model values below. Define any physical scale so the function receives the intended dimensionless argument.

Enter −1.5, −0.75, 0.75, 1.5, 3 in A2:A6 and fill =COTH(A2) down. Expected results are approximately −1.104791393, −1.574433834, 1.574433834, 1.104791393, 1.004969823.

The mathematical magnitude exceeds one for finite nonzero real inputs and approaches one as the input magnitude grows. A native floating-point result can round toward a limiting value, so that ideal bound is not a universal stored-precision certificate.

For input 0.75, both =1/TANH(0.75) and =COSH(0.75)/SINH(0.75) are expected to give 1.574433834. TANH is the reciprocal partner, not TAN.

Reversing COSH/SINH would yield hyperbolic tangent instead. The two results have different magnitudes and roles even though the same pair of component functions is used.

Define the fictional response y=2*COTH(x/3) for positive x. Put 1.5, 3, 6 in D2:D4 and fill:

=2*COTH(D2/3)

Expected responses are approximately 4.327906827, 2.626070571, 2.074629441. The outer factor scales the response, while the divisor changes how the source enters the function. Neither parameter should be moved casually through a nonlinear function.

The positive branch approaches two from above as x grows. It does not reach exactly two at an ordinary finite real x in the mathematical model.

With the unrounded response in E2, =3*ACOTH(E2/2) is expected to reconstruct the original positive x. For the three controls, the reconstructed sequence is approximately 1.5, 3, 6.

ACOTH requires an input outside the inclusive −1-to-1 interval. A rounded response of exactly two would create an excluded inverse ratio of one and lose the original finite input information.

The source’s broad real-input wording must be read alongside its explicit zero-error example. Do not replace a zero with a tiny number without a justified model: COTH grows strongly near the singularity and changes sign across it.

Keep raw inputs and use the documented zero control separately from moderate nonzero calculations. Direct exponential expansions may overflow even when a ratio is near its limiting value. This guide does not establish extreme native accuracy, every text coercion or a general client-wide magnitude limit.

• Google COTH help.
• Editorial testing standards.

Inputs are fictional. Approximate numerical answers are independently calculated mathematical expectations rounded to nine places, not captured native results. Complete coercion, extreme-magnitude and cross-client behavior remain untested.