COSH Function in Zoho Sheet: Hyperbolic Cosine and Catenary Examples

Hyperbolic cosine is written mathematically as cosh(x) = (e^x + e^(-x))/2. It creates a smooth, symmetric curve that reaches its minimum value of 1 when x is zero and grows as the absolute value of x increases. This is useful for illustrating catenary profiles such as an ideal hanging chain, studying hyperbolic geometry, and checking relationships among functions like SINH and TANH.

Unlike COS(x), hyperbolic cosine does not oscillate between -1 and 1. For every finite real x, the mathematical output cosh(x) is at least 1. It has the same value for a positive and corresponding negative input: cosh(-x) = cosh(x). This even-function symmetry is a strong diagnostic when reviewing a worksheet’s results.

Zoho’s public help gives the following syntax:

COSH(number)

A direct formula is =COSH(0.5); a reusable cell reference is =COSH(A2). There is no second parameter to choose degrees, a different logarithmic base, or displayed precision. Change number formatting to control visible decimals without altering the formula itself.

The mathematical function accepts all real inputs, but Zoho’s documentation does not specify a maximum finite value, behaviour for every text input, or exact overflow responses. No unsupported first-release date is assumed; online, mobile, and desktop versions may have different formula-entry conventions depending on edition and file type.

Enter the ten values below in A2:A11 (header x in A1). The same inputs are used in the companion SINH and TANH guides so that numerical identities can be checked on one small dataset.

The results are independent mathematical expectations. They are not screenshots or recorded spreadsheet executions, and minor differences in the last decimal may occur with different calculation and display precision.

Click B2 and enter:

=COSH(A2)

Copy or fill the formula down to B11. When A2 is -3, the expected result is 10.067661996. When A6 is 0, the expected result is exactly 1. When A7 is 0.5, the expected result is 1.127625965. Using a cell reference rather than a hard-coded value makes the table reusable for a modelling worksheet.

To check whether opposite signs produce the same result, place the following formula in an empty cell:

=COSH(-A7)-COSH(A7)

With A7 = 0.5, the expected difference is zero. This is based on the even-function identity, not an observed Zoho computation. If you later test a large input, compare using an absolute or relative tolerance rather than demanding identical displayed strings.

With an input of 1, enter =COSH(1). Expected: 1.543080635. The ordinary circular cosine COS(1) is about 0.540302306 when one radian is supplied. Both names contain the letters COS, but only COSH models the hyperbolic relationship.

You can also check the definition with the documented EXP function:

=(EXP(0.5)+EXP(-0.5))/2

Expected: 1.127625965, matching COSH(0.5). The two calculation routes need not agree to every last machine bit in a spreadsheet; evaluating exponentials then subtracting or adding values can introduce floating-point differences. Use approximately equal comparisons, particularly for values with large magnitude.

In an idealized catenary model, a freely hanging uniform chain in a vertical plane under its own weight can be represented by:

y(x) = a * COSH(x / a)

Here a is a positive scale parameter in distance units, and x is horizontal displacement from the model’s lowest point. If you want height above the minimum, subtract the height at x = 0:

rise(x) = a * (COSH(x / a) - 1)

Suppose a = 10 and x = 5 in the same units. Type:

=10*(COSH(5/10)-1)

Expected rise: approximately 1.276259652 units. At x = -5, the rise is the same. At x = 10, =10*(COSH(10/10)-1) has an expected rise of about 5.430806348. These values can populate a simple curve profile in a teaching spreadsheet. They are model outputs only, not a safe cable-design specification: real engineering requires geometry, material limits, load distribution, anchoring and professional checks.

For a parameterized design sandbox, put a in G2, x in G3, and calculate:

=G2*(COSH(G3/G2)-1)

If G2 = 10 and G3 = 5, expect 1.276259652. Validate that G2 is positive before entering the calculation; division by zero when a = 0 is mathematically invalid for this model, even though it is not an error condition of COSH itself.

For each real x, COSH(x)^2 - SINH(x)^2 = 1. For an independent native test, enter:

=COSH(A7)^2-SINH(A7)^2

Expected: 1 for A7 = 0.5. This relationship is an effective way to catch a bad input reference or an accidental swap of SINH with SIN. At very large magnitudes, subtracting two large nearly equal squared terms can be numerically unstable; a failed high-magnitude identity check does not automatically mean the mathematical functions themselves are wrong.

You can also recover the nonnegative original input magnitude from COSH using inverse hyperbolic cosine: mathematically, ACOSH(COSH(x)) = ABS(x) for real x. For x = -0.5, the expected inverse is +0.5, not -0.5. That loss of sign is inherent because COSH is even, and it matters in inverse-model calculations.

You expect a negative result. For real input, COSH is never negative. Check whether you accidentally intended SINH, a negative multiplier outside COSH, or a circular COS function.

COSH(0) does not display as 1. Check that the value is genuinely numeric zero and not text such as "0 ", and that the formula references the intended cell. Zoho’s generic error guidance includes #NAME! for invalid names, #VALUE! for invalid argument types, and #REF! for broken references; those are documented broad categories, not native tests of the examples here.

An imported decimal is rejected. Data imported from other locales may use decimal commas, text values, or unexpected separators. Start with =COSH(0.5) in a fresh sheet and compare to the value above, then investigate the imported field. Do not assume a particular decimal convention across every Zoho edition or file type.

Huge input produces an error or a distorted value. Exponential functions grow rapidly and finite numeric storage can overflow. Zoho’s public COSH page does not provide a complete numerical limit or an observed specific overflow error for this guide. Use smaller test values and native execution before making implementation claims.

COSH and COS differ. This difference is expected and is often the root cause of a migrated-formula error. COS takes a circular angle in radians, whereas COSH takes a real number as a hyperbolic-function argument. Adding a RADIANS conversion may make sense only if you have a domain-specific model that explicitly calls for it; it is not a universal fix.

Microsoft Excel and Google Sheets have COSH functions with the same fundamental mathematical definition. Both vendor help systems independently describe them as operating on real numeric values. This does not establish that all three programs have matching input coercion rules, error messages, floating-point precision, blank-cell handling, or maximum ranges.

Zoho’s September 2026 change notes describe Excel-native syntax by default for offline files in beta desktop apps on Windows and macOS. Because COSH takes only one argument, differences in multi-argument separators are irrelevant to the basic =COSH(A2) example but can matter in surrounding formulas. Zoho’s online/mobile/account availability or first release of COSH cannot be inferred from this announcement.

Source check: October 12, 2026 UTC. Official Zoho documentation: reviewed. Independent calculations: completed using a separate mathematical implementation. Actual native Zoho calculation: NOT RUN. Native error testing: NOT RUN. Screenshots: NONE.

• SINH in Zoho Sheet — slope-related hyperbolic sine.
• TANH in Zoho Sheet — bounded ratio of hyperbolic sine and cosine.
• ACOSH in Zoho Sheet — inverse for inputs >= 1.
• EXP in Zoho Sheet — the exponential representation.
• Zoho COSH official reference.
• Zoho What’s New: September 2026 desktop support.
• Excel COSH documentation.
• Google Sheets COSH documentation.