SECH in Excel: Build a Symmetric Hyperbolic Peak

• 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 SECH reference. Documentation and independently calculated examples; no native Excel execution or broader platform certification.

=SECH(number)

The argument is required. Microsoft describes it as a radian hyperbolic angle, requires absolute magnitude below 2^27, and documents #NUM! outside that bound and #VALUE! for nonnumeric input.

The help’s sample descriptions label direct SECH(45) and SECH(30) calls as degree angles, although their tiny displayed results correspond to direct numeric inputs. A source angle in degrees needs its stated conversion. A dimensionless model argument, as used below, should be supplied as defined rather than converted automatically.

Enter −2, −1, 0, 1, 2 in A2:A6. Fill =SECH(A2) down in B2:B6. Expected values are approximately 0.265802229, 0.648054274, 1, 0.648054274, 0.265802229.

Opposite inputs produce the same result because COSH is even. The ideal real range is positive and at most one, with the maximum at zero. Extreme native underflow behavior is a separate question and is not certified by that mathematical description.

=1/COSH(1)
=SECH(1)
=SECH(LN(2))

The first two expected outputs are approximately 0.648054274. The third is 0.8, because COSH(LN(2))=(2+1/2)/2=1.25. COSH supplies the reciprocal partner.

Unlike CSCH, SECH has no real zero-input singularity: COSH(0)=1. Do not reject zero merely because another reciprocal hyperbolic function is undefined there.

Define a fictional symmetric response y=12*SECH(x/3). Enter −6, −3, 0, 3, 6 in D2:D6 and fill:

=12*SECH(D2/3)

Expected values are approximately 3.189626746, 7.776651284, 12, 7.776651284, 3.189626746. The multiplier fixes the peak height; the divisor changes the horizontal scale. Keep both parameters labeled when adapting the example.

This is a selected mathematical curve, not evidence that a measured dataset follows it. Preserve source coordinates rather than inferring their signs from equal response values.

Half of the peak is 6, so SECH(x/3)=0.5 and COSH(x/3)=2. The nonnegative solution is:

=3*ACOSH(2)

Expected result: approximately 3.950873691. Its negative is the second half-height position. Substituting either unrounded value into the model gives approximately 6.

ACOSH selects a nonnegative branch, so the second sign must come from the symmetric model. It cannot be recovered from the response alone.

Use the identity =SECH(1)^2+TANH(1)^2, expected to give approximately 1, as a moderate-input check of the correct hyperbolic family. A circular secant or tangent would describe a different curve.

Keep raw inputs and intermediate precision. Very small outputs, rounded endpoints and expanded exponential expressions can lose information at extremes. The documented input limit does not promise uniform relative accuracy across that entire interval.

• Microsoft SECH 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.