ACOSH in Excel: Invert Hyperbolic Cosine with a Nonnegative Branch

• Introduced: First introduction version is not established by the official history reviewed.
• 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 ACOSH reference. Documentation and independently calculated examples; no native Excel execution or broader platform certification.

=ACOSH(number)

The argument is required and must be a real number greater than or equal to 1. The equality matters: one is a valid endpoint. An input of 0.5 is outside the real domain, rather than a small valid result.

The inverse selects the nonnegative branch. COSH is even, so opposite source inputs have the same hyperbolic cosine. An inverse cannot recover a sign that the forward value no longer contains.

Enter 1, 1.5, 2, 3, 5 in A2:A6. Put =ACOSH(A2) in B2 and fill down. Expected results are approximately 0, 0.962423650, 1.316957897, 1.762747174, 2.292431670.

These are model values, not circular angles to convert automatically into degrees. Keep the original COSH argument’s meaning and units available when working backward.

=ACOSH(COSH(1))
=ACOSH(COSH(-1))

Both expected outputs are approximately 1. The second does not return −1. The source help’s general inverse-composition sentence must be read with this principal-branch restriction: for real x, ACOSH(COSH(x)) reconstructs its magnitude.

For a permitted value such as 3, the other direction =COSH(ACOSH(3)) reconstructs approximately 3. The two composition orders answer different questions because the forward COSH is not one-to-one over all real inputs.

Suppose a fictional response is defined as y=5*(COSH(x/2)-1), and the desired solution is the nonnegative x. For target y values 0, 5, 10 in D2:D4, use:

=2*ACOSH(1+D2/5)

Expected nonnegative solutions are approximately 0, 2.633915794, 3.525494348. Substitution into the original expression reconstructs 0, 5, 10 within normal numerical tolerance.

For a positive target, the negative of each nonzero solution also produces the same response. Choose the branch from the application, not merely from the first value a function returns. This example explicitly chooses x≥0.

A negative y in this model makes 1+y/5 less than one, outside ACOSH’s real domain. For ordinary scalar numeric inputs, an explicit policy is:

=IF(D2>=0,2*ACOSH(1+D2/5),"Review target")

A target of −1 is intended to produce Review target, while zero produces zero. This policy does not claim universal lazy evaluation in every nested-array formula.

Near the endpoint one, small source differences can produce meaningful changes in the inverse. Inspect the stored input before rounding or clamping it to one. A silent clamp can conceal a target that the stated model cannot achieve.

The official page establishes the domain but does not supply a complete native error/coercion matrix. This guide does not borrow one from another spreadsheet product. Keep the selected branch and source precision visible in the workbook.

• Microsoft ACOSH 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.