How the cosh() function works in SQLite?

While most database professionals are familiar with basic trigonometric functions, the hyperbolic cosine function cosh() remains a hidden gem in SQLite’s mathematical toolkit. This powerful function extends beyond ordinary trigonometry into areas like advanced physics, engineering calculations, and financial modeling. Unlike its circular cousin cos(), cosh() operates on hyperbolas rather than circles, making it indispensable for certain specialized calculations. Whether you’re modeling catenary curves, calculating relativistic effects, or working with exponential growth patterns, understanding cosh() can give you an edge in solving complex mathematical problems directly within your SQL queries.

The Nature of Hyperbolic Cosine

The cosh() function computes the hyperbolic cosine of a number, which is defined mathematically as (eˣ + e⁻ˣ)/2. This gives it distinctive properties compared to the standard cosine function:

SELECT cosh(0);   -- Returns 1.0 (as e⁰ + e⁻⁰ = 1 + 1 = 2, then divided by 2)
SELECT cosh(1);   -- Approximately 1.54308
SELECT cosh(-1);  -- Also approximately 1.54308 (cosh is symmetric)

Notice how cosh() always returns values ≥ 1 and produces identical results for positive and negative inputs, unlike the oscillating cos() function.

Modeling Real-World Curves and Shapes

One of the most practical applications of cosh() is modeling natural hanging shapes like suspension bridge cables or hanging chains (catenaries):

-- Catenary curve calculation (y = a*cosh(x/a))
SELECT
    x_position,
    10 * cosh(x_position / 10) AS cable_height
FROM bridge_points;

This calculates the vertical position of points along a hanging cable with scaling factor 10.

Special Relativity Calculations

In physics applications, cosh() appears in relativistic velocity addition formulas:

-- Relativistic velocity composition
SELECT (v1 + v2) / (1 + (v1 * v2) / (c * c)) AS classical_sum,
       c * tanh(acosh(gamma1) + acosh(gamma2)) AS relativistic_sum
FROM velocities
WHERE gamma1 = cosh(acosh_v1) AND gamma2 = cosh(acosh_v2);

While SQLite isn’t a physics simulation tool, this demonstrates cosh()’s role in scientific computing.

Financial Mathematics Applications

Hyperbolic functions appear in certain financial models, particularly those involving continuous growth:

-- Continuous growth comparison
SELECT
    time_period,
    exp(growth_rate * time_period) AS exp_growth,
    cosh(growth_rate * time_period) + sinh(growth_rate * time_period) AS hyperbolic_growth
FROM investment_periods;

This shows the relationship between exponential and hyperbolic growth models.

Combining with Other Hyperbolic Functions

cosh() becomes particularly powerful when used with its hyperbolic counterparts:

-- Identity verification
SELECT cosh(1.5) * cosh(1.5) - sinh(1.5) * sinh(1.5);  -- Returns 1.0

-- Calculating hyperbolic secant
SELECT 1 / cosh(input_value) AS sech_value FROM calculations;

These examples demonstrate mathematical relationships between hyperbolic functions.

Handling Edge Cases and Special Values

Understanding cosh()’s behavior at boundaries is crucial for robust implementations:

SELECT cosh(NULL);   -- Returns NULL
SELECT cosh(1e308);  -- May return Infinity or cause overflow
SELECT cosh(-1e308); -- Same behavior as positive large numbers
SELECT cosh(0.0);    -- Returns exactly 1.0

The function is well-behaved except at extreme values where floating-point limitations apply.

Performance Considerations

When working with cosh(), keep these performance aspects in mind:

1. Hyperbolic function calculations are more computationally intensive than basic arithmetic
2. For repeated calculations with the same input, consider storing results in temporary tables
3. Complex expressions combining multiple hyperbolic functions can often be simplified mathematically
4. SQLite’s implementation may vary in precision across different platforms

Practical Implementation Notes

While cosh() is available in standard SQLite, some considerations apply:

-- Check for function availability
SELECT cosh(1) FROM sqlite_master WHERE type='function';

-- Alternative implementation if unavailable
SELECT (exp(1.0) + exp(-1.0)) / 2.0 AS manual_cosh;

Some lightweight SQLite builds might exclude mathematical functions.

Conclusion

SQLite’s cosh() function provides a gateway to advanced mathematical operations that many database professionals overlook. From modeling physical structures to solving engineering problems and analyzing growth patterns, this hyperbolic function offers capabilities far beyond ordinary SQL arithmetic. While not needed for everyday database operations, cosh() proves invaluable when working with specialized mathematical models directly in your SQL environment. Its presence in SQLite demonstrates the database engine’s surprising mathematical sophistication. Next time you encounter a problem involving hyperbolic relationships, exponential growth, or certain physical models, remember that cosh() might provide the elegant solution you need - all without leaving your SQL query.