ACOSH in Google Sheets: Solve a Hyperbolic Cosine Model with a Chosen Branch
Use Sheets ACOSH with an inclusive lower bound, nonnegative inverse examples and an explicit symmetric-model solution.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: ACOSH returns inverse hyperbolic cosine for real inputs at least one. =ACOSH(1.25) is expected to give approximately 0.693147181, while =ACOSH(1) gives 0.
Availability and version history
- 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 ACOSH reference. Documentation and independent calculations, without native Sheets execution.
Syntax and domain
=ACOSH(value)
Google explicitly requires value≥1. The boundary one is included; a value such as 0.5 is outside the real domain. The one input is a hyperbolic cosine value rather than a circular degree angle.
The real inverse uses a nonnegative branch. Since COSH gives the same value for opposite real inputs, the output alone cannot recover the source sign. Keep branch choice as part of the worksheet model.
Example 1: compare the endpoint with larger values
Enter 1, 1.25, 2.5, 4, 10 in A2:A6 and fill =ACOSH(A2) down. Expected results are approximately 0, 0.693147181, 1.566799237, 2.063437069, 2.993222846.
These are ordinary real mathematical expectations. No additional degree conversion is required by the model. The help’s general complex-number note is scoped here to this real-valued function, not generalized across every separate Sheets function.
Example 2: make sign loss visible
=ACOSH(COSH(2))
=ACOSH(COSH(-2))
Both expected results are approximately 2. The second does not reconstruct −2. In the other direction, =COSH(ACOSH(4)) reconstructs approximately 4, because four is a permitted inverse input.
An inverse composition can therefore recover magnitude without recovering sign. Store the original sign or an explicit branch indicator if both are relevant.
Advanced example: invert a shifted symmetric curve
Define a fictional response y=3*(COSH(x/4)-1) and request the nonnegative x solution. For y values 0, 3, 9 in D2:D4:
=4*ACOSH(1+D2/3)
Expected solutions are approximately 0, 5.267831588, 8.253748276. Substituting each unrounded solution into the original response reconstructs 0, 3, 9.
For each positive y, the negative of the nonzero solution is another solution of the same symmetric equation. ACOSH returns the selected nonnegative value; it does not decide which physical direction an application intended.
Advanced example: show invalid targets before calculating
For ordinary numeric cells, the scalar policy =IF(D2>=0,4*ACOSH(1+D2/3),"Review target") is intended to return Review target for y=−3 and 0 for y=0.
Negative targets would make the ACOSH input smaller than one. The label makes that incompatibility visible rather than clamping the input to an endpoint. This controlled IF example does not certify lazy evaluation in every array expression.
Precision and interpretation
Close to input one, small differences can affect the inverse appreciably. Inspect the stored value before rounding it to one, and retain the original response for validation. A mathematically permitted inverse still does not establish that the chosen curve fits real measurements.
The short Google help does not give a full native error-code or coercion matrix. These moderate examples verify the intended mathematics, while deployment-specific boundary tests remain separate.
References and verification
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.