Calculate hyperbolic angles using cosh and log

Introduction to hyperbolic angles and hyperbolic functions

Hyperbolic angle $\theta$ is the independent variable that defines a hyperbolic function, similar to the angle in a trigonometric function. The hyperbolic cosine function is defined as:

According to the definition of the inverse function, if a hyperbolic cosine value is given $y = \cosh \theta$ , then the corresponding hyperbolic angle can be obtained:

here, $\cosh^{-1}$ is an inverse hyperbolic cosine function, and the log function corresponds to natural logarithm.

Implementation of calculating hyperbolic angles in PHP

PHP 7.2 and above have built-in hyperbolic functions and inverse hyperbolic functions, such as cosh() and acosh() , but for a better understanding, we manually use the combination of cosh() and log() to calculate.

Here is the sample code:

 <?php
// Enter the hyperbolic cosine value
$y = 2.5;

// Calculate hyperbolic angles θ = acosh(y) = ln(y + sqrt(y^2 - 1))
if ($y < 1) {
    echo "The input value must be greater than or equal to1，There is a corresponding hyperbolic angle。";
} else {
    $theta = log($y + sqrt($y * $y - 1));
    echo "Hyperbolic cosine value y = $y The corresponding hyperbolic angle θ = $theta";
}
?>

Code description:

• Since the definition of hyperbolic cosine is $[1, \infty)$ , the input value must be greater than or equal to 1.

• Use log() to calculate natural logarithm.

• Use sqrt() to calculate the square root.

• Finally got $\theta$ is the corresponding hyperbolic angle.

Practical application examples

Suppose there is a physical model that the corresponding hyperbolic cosine value of a certain quantity is 3, and you want to know the corresponding hyperbolic angle:

 <?php
$cosh_value = 3;

if ($cosh_value < 1) {
    echo "Invalid input";
} else {
    $angle = log($cosh_value + sqrt($cosh_value * $cosh_value - 1));
    echo "Hyperbolic angle is: " . $angle;
}
?>

Output:

 Hyperbolic angle is: 1.76274717403909

Summarize

Through PHP's log() and sqrt() functions, we can easily implement the calculation of hyperbolic angles and make full use of the properties of hyperbolic cosine functions. Although PHP 7.2 and above directly support the acosh() function, understanding its mathematical principles can help more flexible application and extension.

If you need to access more mathematical functions documentation, you can refer to the relevant resources of gitbox.net.