C Math cosh() Function

What You’ll Learn

The cosh() function computes the hyperbolic cosine of a real number. Built from exponentials—(ex + e−x) / 2—it always returns a value ≥ 1 and appears in physics, engineering, and special relativity formulas.

Definition and Usage

Hyperbolic cosine is defined as cosh(x) = (ex + e−x) / 2, where e is Euler’s number (about 2.718). Unlike circular cos(), which oscillates between −1 and 1, cosh(x) grows smoothly and is always at least 1.

The input x is any real number—not a degree or radian angle in the circular sense. Think of it as a hyperbolic parameter. At x = 0, both exponentials contribute equally and cosh(0) = 1.

📝 Syntax

Standard C declaration:

double cosh(double x);

Related variants

float coshf(float x);           /* <math.h> */
long double coshl(long double x); /* <math.h> */

Parameters

• x — any real number (hyperbolic argument).

Return Value

• Hyperbolic cosine of x, always ≥ 1.
• cosh(0) returns 1.0.
• cosh(1.5) returns about 2.352.
• For very large |x|, result may overflow to infinity.

Headers and linking

• #include <math.h>
• Compile: gcc cosh.c -std=c11 -o cosh -lm

Examples Gallery

Compile every example with gcc file.c -std=c11 -o out -lm. Hyperbolic functions pair naturally with exp() and sinh().

#include <math.h>
#include <stdio.h>

int main(void) {
    double x = 1.5;
    double result = cosh(x);

    printf("cosh(%.1f) = %.6f\n", x, result);

    return 0;
}

cosh(1.5) = 2.352410

#include <math.h>
#include <stdio.h>

int main(void) {
    printf("cosh(0) = %.1f\n", cosh(0.0));

    return 0;
}

cosh(0) = 1.0

#include <math.h>
#include <stdio.h>

int main(void) {
    double x = 1.5;

    printf("cosh( %.1f) = %.6f\n",  x, cosh(x));
    printf("cosh(%.1f) = %.6f\n", -x, cosh(-x));

    return 0;
}

cosh( 1.5) = 2.352410
cosh(-1.5) = 2.352410

#include <math.h>
#include <stdio.h>

int main(void) {
    double x = 2.0;
    double manual = (exp(x) + exp(-x)) / 2.0;
    double library = cosh(x);

    printf("x = %.1f\n", x);
    printf("Manual:   %.6f\n", manual);
    printf("cosh():   %.6f\n", library);

    return 0;
}

x = 2.0
Manual:   3.762196
cosh():   3.762196

#define _USE_MATH_DEFINES
#include <math.h>
#include <stdio.h>

int main(void) {
    double vals[] = { 0.0, 1.0, 1.5, M_PI / 3 };
    size_t i;

    printf("    x     cosh(x)   cos(x)\n");
    for (i = 0; i < 4; i++) {
        double x = vals[i];
        printf("%5.2f  %8.4f  %8.4f\n", x, cosh(x), cos(x));
    }

    return 0;
}

    x     cosh(x)   cos(x)
 0.00    1.0000    1.0000
 1.00    1.5431    0.5403
 1.50    2.3524    0.0707
 1.05    1.6003    0.5000

🚀 Common Use Cases

• Catenary curves — shape of a hanging cable: y = a · cosh(x/a).
• Special relativity — Lorentz factor involves cosh in rapidity formulations.
• Signal processing — hyperbolic functions in filter design.
• Heat and diffusion — solutions to differential equations.
• Pair with sinh() — hyperbolic identities like cosh²(x) − sinh²(x) = 1.

📝 Notes

• Defined as (ex + e−x) / 2.
• Domain: all real numbers (not just 0 to ∞).
• Range: ≥ 1; minimum at cosh(0) = 1.
• Even function: cosh(-x) == cosh(x).
• Not the same as cos()—no degree/radian conversion needed.
• Very large |x| may cause overflow; the library handles this gracefully.

⚡ Optimization

cosh() is implemented in the platform math library with careful handling of overflow. Avoid rewriting it as (exp(x) + exp(-x)) / 2 in hot loops—direct exp calls can overflow sooner. Use coshf when float precision is sufficient to save memory bandwidth.

💡 Best Practices