You probably want the trigger to fire when the player is present, so group the trigger to the player .... yes, just as if the trigger was a unit. Now look in the trigger, you will find new options in the on activation box. Select the one you want.
Now, the condition field
Condition: this and ObjComplete
The trigger will only fire when the player is in the trigger area and the variable ObjComplete is true.
You could split up the objective variables so you could have
this and Obj_1_complete and Obj_2_complete