If I understand your problem correctly:
You want to generate a random point (x,y) for a player that owns the chunk of circle between 0deg-15deg and your circle radius is 1 unit...
Randomly generate a vector as a length of 0-1 and an angle between 0-15deg.
The end point of that vector will be
x = L * cos(angle)
y = L * sin(angle)
You need to watch the convention of the sin/cos function you are using with regards to degrees vs radians.
Note that this will create a higher density of points close to the center of the disc. If you'd prefer a uniform distribution of points in the xy-plane, either generate (x, y)-pairs until one point falls within the circular sector of the player of interest, or scale the points by sqrt(L):
x = sqrt(L) * cos(angle); y = sqrt(L) * sin(angle);
Read up on polar integration for more info.