Say the quadrants are numbered as such:

\1/2x3/4\ 

Create 2 variables: dx and dy (meaning delta x and delta y)

dx = mousex - playerx //Calculate the difference in xdy = mousey - playery //Same for yif(abs(dx) > abs(dy)) { //Does the dx have greater magnitude than the dy?    if(dx > 0) {         //Cursor is in quadrant 3    } else {        //Cursor is in quadrant 2    }} else {    if(dy > 0) {        //Cursor is in quadrant 1    } else {        //Cursor is in quadrant 4    }} 

Let me know if you need an explination of what is going on.

The way it works is you make dx and dy hold the distance from the player to the cursor in the x and y directions. So if you take the absolute value of this distance, you can compare them to see whether the cursor is further from the player in the x direction or the y direction. If you look at your drawing, you can see that if the x distance is greater than the y distance, the cursor must be in quadrants 2 or 3. Otherwise, it must be in 1 or 4. Then, all you have to do is find the sign of dx or dy to find exactly which quadrant it is in.

Are you converting the coordinates so that the player and mouse are represented using the same scale? This is necesary in order for dx and dy to provide useful information.