C Math atan() Function

What You’ll Learn

The atan() function computes the arc tangent (inverse tangent) of a number. Given a slope or tangent ratio, it returns the angle in radians whose tangent equals that value—ideal for line angles, ramps, and graphics.

Definition and Usage

atan (arc tangent) answers: “What angle has this tangent?” If tan(θ) = x, then atan(x) = θ (in radians, principal value).

On a right triangle, tangent is opposite ÷ adjacent (rise over run). If a hill rises 3 units for every 4 units forward, the slope is 3/4 and atan(3.0/4.0) gives the incline angle.

📝 Syntax

Standard C declaration:

double atan(double x);

Related variants

float atanf(float x);           /* <math.h> */
long double atanl(long double x); /* <math.h> */

Parameters

• x — tangent value (slope ratio); any real number.

Return Value

• Angle in radians in the range [−π/2, π/2].
• atan(0) returns 0; atan(1) returns π/4.
• atan(±∞) approaches ±π/2 (infinity handled by the library).

Headers and linking

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

Examples Gallery

Compile every example with gcc file.c -std=c11 -o out -lm. Define _USE_MATH_DEFINES before <math.h> on MSVC for M_PI.

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

int main(void) {
    double x = 0.5;
    double rad = atan(x);
    double deg = rad * 180.0 / M_PI;

    printf("The arctangent of %.2f is %.2f radians or %.2f degrees.\n",
           x, rad, deg);

    return 0;
}

The arctangent of 0.50 is 0.46 radians or 26.57 degrees.

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

int main(void) {
    printf("atan( 0) = %.6f radians\n", atan(0.0));
    printf("atan( 1) = %.6f radians\n", atan(1.0));
    printf("atan(-1) = %.6f radians\n", atan(-1.0));

    return 0;
}

atan( 0) = 0.000000 radians
atan( 1) = 0.785398 radians
atan(-1) = -0.785398 radians

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

int main(void) {
    double rise = 3.0;
    double run  = 4.0;
    double slope = rise / run;
    double angle_deg = atan(slope) * 180.0 / M_PI;

    printf("Slope = %.0f / %.0f = %.2f\n", rise, run, slope);
    printf("Incline angle = %.1f degrees\n", angle_deg);

    return 0;
}

Slope = 3 / 4 = 0.75
Incline angle = 36.9 degrees

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

int main(void) {
    double values[] = { -2.0, -0.5, 0.0, 0.5, 2.0 };
    size_t i;

    for (i = 0; i < sizeof values / sizeof values[0]; i++) {
        double x = values[i];
        double angle = atan(x);
        double back = tan(angle);

        printf("x=%5.1f  atan(x)=%.4f  tan(atan(x))=%.6f\n",
               x, angle, back);
    }

    return 0;
}

x= -2.0  atan(x)=-1.1071  tan(atan(x))=-2.000000
x= -0.5  atan(x)=-0.4636  tan(atan(x))=-0.500000
x=  0.0  atan(x)=0.0000  tan(atan(x))=0.000000
x=  0.5  atan(x)=0.4636  tan(atan(x))=0.500000
x=  2.0  atan(x)=1.1071  tan(atan(x))=2.000000

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

int main(void) {
  /* Both points have slope y/x = -0.75 */
  double x1 =  4.0, y1 = -3.0;
  double x2 = -4.0, y2 =  3.0;

  printf("Point A (%.0f, %.0f):\n", x1, y1);
  printf("  atan(y/x)  = %.2f rad\n", atan(y1 / x1));
  printf("  atan2(y,x) = %.2f rad\n", atan2(y1, x1));

  printf("Point B (%.0f, %.0f):\n", x2, y2);
  printf("  atan(y/x)  = %.2f rad\n", atan(y2 / x2));
  printf("  atan2(y,x) = %.2f rad\n", atan2(y2, x2));

  return 0;
}

Point A (4, -3):
  atan(y/x)  = -0.64 rad
  atan2(y,x) = -0.64 rad
Point B (-4, 3):
  atan(y/x)  = -0.64 rad
  atan2(y,x) = 2.50 rad

🚀 Common Use Cases

• Line and slope angles — convert rise/run into an incline angle.
• 2D graphics — rotation from a known tangent when quadrant is fixed.
• Robotics — heading adjustments from velocity component ratios.
• Geometry — find angles in right triangles via opposite/adjacent.
• Pair with tan() — forward and inverse tangent operations.

📝 Notes

• Input x is a tangent ratio, not an angle in radians.
• Output is in [−π/2, π/2] radians—limited to one half of the circle.
• For points in all four quadrants, use atan2(y, x) instead of atan(y/x).
• Link with -lm when using GCC/Clang on Unix-like systems.
• Use atanf or atanl for float or long double data.

⚡ Optimization

atan() is implemented in the platform math library and is already optimized. Prefer atan2(y, x) over atan(y/x) when you have separate coordinates—it avoids a division and handles x = 0 safely while preserving quadrant information.

💡 Best Practices