Understanding SQLite’s COSH() Function

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The COSH() function in SQLite calculates the hyperbolic cosine of a number, which is similar to the regular cosine function, but for hyperbolic geometry.

Syntax

COSH(x)

Where x is the input value (in radians) for which you want to compute the hyperbolic cosine.

Hyperbolic Cosine Formula

The hyperbolic cosine of a number x is given by the formula: 

cosh(x) = (e^x + e^(-x)) / 2

Where:

  • e is Euler’s number, approximately 2.71828.
  • x is the input value in radians.

Example 1

Here’s a basic example to demonstrate COSH():

SELECT COSH(0);

Output:

1.0

The hyperbolic cosine of 0 is 1 because cosh(0) = (e0 + e0) / 2 = (1 + 1) / 2 = 1.

Example 2

Let’s do an example that uses COSH() against data in a table:

CREATE TABLE t1 (x REAL);

INSERT INTO t1 (x) VALUES (0), (1), (-1), (2), (-2);

SELECT x, COSH(x) AS hyperbolic_cosine
FROM t1;

Output:

x     hyperbolic_cosine
----  -----------------
0.0   1.0              
1.0   1.54308063481524 
-1.0  1.54308063481524 
2.0   3.76219569108363 
-2.0  3.76219569108363

Explanation:

  • COSH(0) is 1 because for x = 0, the hyperbolic cosine is always 1 (as shown in the previous formula).
  • COSH(1) and COSH(-1) give the same result because the hyperbolic cosine function is an even function, meaning cosh(x) = cosh(-x).
  • For larger values of x, the hyperbolic cosine increases exponentially due to the contributions of the terms (ex + e-x) / 2, where the term ex dominates as x grows, making cosh(x) grow exponentially.

Summary

  • COSH(x) computes the hyperbolic cosine of the input value x in radians.
  • It uses the formula cosh(x) = (ex + e-x) / 2 to calculate the hyperbolic cosine of x.
  • The function is useful for calculations in hyperbolic geometry and other fields of mathematics where hyperbolic functions are needed.