SECH in Google Sheets: Build a Symmetric Peak and Find Its Half-Height

• 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 SECH reference. Documentation and independent calculations, without native Sheets execution.

=SECH(value)

Google describes a real input value. No circular-angle conversion is implied by the dimensionless model controls in this guide. SECH is different from SEC, whose argument is a circular radian angle.

COSH never reaches zero for a real input, so SECH has no real zero-input singularity. Its mathematical range is positive and at most one. Native underflow at extreme magnitudes is a different question and is not certified by that ideal range.

Enter −3, −1.5, 0, 1.5, 3 in A2:A6 and fill =SECH(A2) down. Expected outputs are approximately 0.099327927, 0.425096035, 1, 0.425096035, 0.099327927.

The equal results at opposite inputs show the lost sign information. An output value alone cannot tell whether the source coordinate was positive or negative. Keep the original coordinate column.

=SECH(LN(3))
=1/COSH(LN(3))

Both expected results are 0.6, because COSH(LN(3))=(3+1/3)/2=5/3. COSH supplies the reciprocal partner.

At zero, COSH is one, which explains SECH’s maximum of one. This is different from CSCH’s zero denominator and documented error. Similar function names do not establish identical domains.

Define a fictional response y=8*SECH(x/2). Put −4, −2, 0, 2, 4 in D2:D6 and fill:

=8*SECH(D2/2)

Expected values are approximately 2.126417831, 5.184434189, 8, 5.184434189, 2.126417831. The multiplier sets the peak at eight; the input divisor sets the horizontal scale.

Changing either parameter changes the model. A symmetric curve that looks smooth is not evidence that it is appropriate for a particular set of observations.

Half-height is four, so the equation becomes COSH(x/2)=2. The nonnegative position is:

=2*ACOSH(2)

Expected value: approximately 2.633915794. Its negative is the second half-height position. Substituting either unrounded value into the scaled SECH formula gives approximately 4.

ACOSH supplies a nonnegative inverse branch. The second sign comes from the model’s symmetry, not from another output cell automatically returned by the function.

At x=0.75, =SECH(0.75)^2+TANH(0.75)^2 is expected to be approximately 1. This identity helps detect a mistaken circular function family on a moderate input.

Keep full intermediate precision and use a justified tolerance rather than exact equality for numerical checks. Very small outputs may lose recoverable information when rounded, while large exponential intermediates can complicate hand-expanded formulas.

The short Google reference does not provide a universal magnitude limit, full type-coercion policy or cross-client numerical certification. These independently calculated controls establish the intended ordinary real mathematics only.

• Google SECH 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.