C++ :: Headers And Objects Files

I have a question regarding how GCC relates a header file and its binary file.

main.c
#include <math.h>
#include <stdio.h>
int main (void){
double x = sqrt (2.0);
printf ("The square root of 2.0 is %f
", x);
return 0;
}

we can compile it by running the line: gcc main.c -lm -o main

My question is: How GCC knows where is the definition of sqrt?

First I was thinking that there was and object file with the name math.o inside libm.a (that is GCC will look for an object file with the same name as the header file), but after running the next line I think my assumption was wrong as there is not such file in libm.a.

nm libm.a | grep math.o
nm: e_acos.o: no symbols
nm: k_cos.o: no symbols
nm: k_sin.o: no symbols

[Code] ....