acosh()

The PHP acosh() function returns the inverse hyperbolic cosine (also called the area hyperbolic cosine) of a number — the value whose hyperbolic cosine equals the given argument. This page covers its syntax, the valid input range, return values, edge cases, and a worked example.

Syntax

acosh(float $num): float

The return value is a float: the inverse hyperbolic cosine of $num, expressed in radians.

What acosh() Does

acosh() is the inverse of cosh(). If cosh($x) gives you $y, then acosh($y) gives you back $x. Mathematically:

acosh(x) = ln(x + sqrt(x*x - 1)),  for x >= 1

Because the hyperbolic cosine never drops below 1, the inverse is only defined for inputs >= 1. Any value below 1 falls outside the function's real domain.

Basic Example

Output:

0.96242365011921

Here we pass 1.5 to acosh() and print the result in radians. You can verify it: cosh(0.96242365011921) returns 1.5.

Input Range and Edge Cases

The domain of acosh() starts at 1. Watch how the boundaries behave:

<?php
echo acosh(1) . "\n";    // smallest valid input
echo acosh(10) . "\n";   // a larger value
echo acosh(0.5) . "\n";  // below the domain
echo acosh(-2) . "\n";   // negative input
?>

Output:

0
2.9932228461264
NAN
NAN

Key points:

• acosh(1) is exactly 0, because cosh(0) is 1.
• Any argument less than 1 (including negatives) returns NAN ("not a number"), since the real result is undefined there.
• Use is_nan() to detect and guard against invalid results before using them.

<?php
$value = 0.5;
$result = acosh($value);

if (is_nan($result)) {
    echo "acosh() is only defined for values >= 1.";
} else {
    echo $result;
}
?>

When Would I Use acosh()?

Hyperbolic functions show up in physics and engineering contexts — for example, the shape of a hanging cable or chain (a catenary), special relativity calculations, and certain signal-processing formulas. Whenever you have a hyperbolic-cosine value and need to recover the original argument, acosh() is the tool.

Related Functions

• cosh() — hyperbolic cosine (the inverse operation).
• asinh() — inverse hyperbolic sine.
• atanh() — inverse hyperbolic tangent.
• acos() — arc cosine (the circular, non-hyperbolic counterpart).

Summary

acosh() computes the inverse hyperbolic cosine of a number in radians. Pass it a value >= 1 to get a real result; anything smaller returns NAN. It pairs with cosh() as its inverse and is most useful in scientific and engineering math.

Practice