C Math sinh() Function

What You’ll Learn

The sinh() function computes the hyperbolic sine of a real number. Built from exponentials—(ex − e−x) / 2—it passes through zero at x = 0 and grows without bound as |x| increases.

Definition and Usage

Hyperbolic sine is defined as sinh(x) = (ex − e−x) / 2, where e is Euler’s number (about 2.718). Unlike circular sin(), which oscillates between −1 and 1, sinh(x) grows smoothly and is unbounded.

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 cancel and sinh(0) = 0.

📝 Syntax

Standard C declaration:

double sinh(double x);

Related variants

float sinhf(float x);             /* <math.h> */
long double sinhl(long double x); /* <math.h> */

Parameters

• x — any real number (hyperbolic argument).

Return Value

• Hyperbolic sine of x as double.
• sinh(0) returns 0.0.
• sinh(1.5) returns about 2.129.
• For very large |x|, result may overflow to ±infinity.

Headers and linking

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

Examples Gallery

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

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

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

    printf("The hyperbolic sine of %.2f is: %.4f\n", x, result);

    return 0;
}

The hyperbolic sine of 1.50 is: 2.1293

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

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

    return 0;
}

sinh(0) = 0.0

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

int main(void) {
    double x = 1.5;

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

    return 0;
}

sinh( 1.5) = 2.129280
sinh(-1.5) = -2.129280

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

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

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

    return 0;
}

x = 2.0
Manual:   3.626860
sinh():   3.626860

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

int main(void) {
    double vals[] = { 0.0, 1.0, 1.5, M_PI / 2 };
    size_t i;

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

    return 0;
}

    x     sinh(x)   sin(x)
 0.00    0.0000    0.0000
 1.00    1.1752    0.8415
 1.50    2.1293    0.9975
 1.57    2.3013    1.0000

🚀 Common Use Cases

• Physics and engineering — solutions to differential equations, heat flow.
• Special relativity — rapidity and Lorentz transformations use hyperbolic functions.
• Signal processing — filter design with hyperbolic kernels.
• Finance — models involving exponential growth and decay.
• Pair with cosh() — identity cosh²(x) − sinh²(x) = 1.

📝 Notes

• Defined as (ex − e−x) / 2.
• Domain: all real numbers.
• sinh(0) = 0; output is unbounded as |x| grows.
• Odd function: sinh(-x) == -sinh(x).
• Not the same as sin()—no degree/radian conversion needed.
• Very large |x| may cause overflow; the library handles this gracefully.

⚡ Optimization

sinh() 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 sinhf when float precision is sufficient to save memory bandwidth.

💡 Best Practices