The C programming atan2 function is defined in the math.h header file and is used to calculate the arctangent of the quotient of two given values, (y/x), in radians. The arctangent function, denoted atan(y/x), returns the angle (in radians) whose tangent is y/x.

The atan2 function takes two arguments of type double, which represent the numerator y and denominator x of the quotient whose arctangent is to be calculated. The function returns a double value, which represents the calculated arctangent in radians.

Here's an example usage of the atan2 function to calculate the arctangent of a given input:

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

int main() {
    double y = 0.5;
    double x = 1.0;
    double arctangent = atan2(y, x);
    
    printf("The arctangent of %f/%f is %f radians\n", y, x, arctangent);
    
    return 0;
}

In this example, the atan2 function is used to calculate the arctangent of the quotient 0.5/1.0, which is stored in the y and x variables, respectively. The calculated arctangent in radians is then stored in the arctangent variable, and printed to the console using the printf function.

Note that the atan2 function returns a value between -π and π radians, inclusive. Also note that the function handles the signs of y and x correctly, so the result will be in the correct quadrant of the unit circle. For example, atan2(1, -1) returns 3π/4 radians, since 1 is in the positive y-axis and -1 is in the negative x-axis.