COTH in Excel: Calculate Signed Hyperbolic Cotangent

• Introduced: Microsoft marks the function as introduced in Excel 2013. Version-marker reference.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official COTH reference. Documentation and independently calculated examples; no native Excel execution or broader platform certification.

=COTH(number)

The argument is required. Microsoft limits its absolute value to less than 2^27, documents #NUM! outside that bound and #VALUE! for nonnumeric input. Those are application constraints in addition to the mathematical definition.

A hyperbolic model should define the argument and its scale. Do not automatically convert a measured quantity into radians just because a related circular function accepts angles. The worked inputs here are specified dimensionless real model values.

Enter −2, −0.5, 0.5, 1, 2 in A2:A6. In B2 enter =COTH(A2) and fill down. Expected outputs are approximately −1.037314721, −2.163953414, 2.163953414, 1.313035285, 1.037314721.

The signs reverse with the input. For ordinary finite nonzero real arguments, the mathematical magnitude exceeds one and approaches one as the input magnitude grows. Stored arithmetic may round toward a limiting value; the ideal inequality does not certify every floating-point result.

At input 1:

=1/TANH(1)
=COSH(1)/SINH(1)
=COTH(1)

Expected results are approximately 1.313035285. TANH is the reciprocal partner, and COSH/SINH express the same ordinary real ratio.

The sign and denominator matter. Reversing the fraction to SINH/COSH produces TANH, approximately 0.761594156, rather than COTH.

For inputs 0.1 and −0.1, expected COTH values are approximately 10.033311132 and −10.033311132. These are large valid ratios on opposite sides of the singularity.

At exactly zero, the real mathematical ratio is undefined because SINH(0)=0. The dedicated COTH page reviewed does not explicitly pin the native zero-input error code. Do not infer a finite origin value from the finite nearby controls, and do not silently replace zero with an arbitrary tiny number.

Suppose the chosen positive-branch model is y=3*COTH(x/2). For target y=6:

=2*ACOTH(6/3)

Expected x is approximately 1.098612289. Substitution into the original model reconstructs approximately 6. For target y=12 the corresponding x is approximately 0.510825624.

ACOTH requires its ratio to have magnitude greater than one. The positive branch therefore cannot reach y=3 at a finite x or a target between zero and three. State that restriction before interpreting a failed inverse.

Direct exponential expansions can create huge intermediates even when the final ratio is near one. An identity is a check for the stated moderate inputs, not a replacement algorithm for every magnitude.

Preserve raw inputs, reject singular cases under a visible policy and keep the model’s scale explicit. Native overflow/underflow details, exact boundary rounding and unusual imported types remain outside these independent tests.

• Microsoft COTH reference.
• Editorial testing standards.

Worked inputs are fictional. Decimal answers are independently calculated mathematical expectations rounded to nine places, not captured native results. Extreme magnitudes, complete coercion matrices and cross-version numerical accuracy remain untested.