The 3" vertical that you plan to use as the sink drain appears to be your main waste stack. There are different kinds of "tee fittings", and the one you show is a Sanitary tee. That is usually used for horizontal to vertical drain connections and can be used there. However, as that is your main waste stack, you will need a separate vent for the sink P-trap. It appears that the line to the left in your picture is a vent line. Just confirm that and you can run a vent line from your sink P-trap over to it. The vent line needs to be vertical, or no more than 45 degrees off vertical, until it reaches 6" above the flood rim of the sink. Then it can be run at a slope of 1/4" per foot. And since the drain connection to the stack doesn't need to be used as a vent, it would be better to use a wye or a combo fitting for the drain connection. For the vent connection, you can use any tee fitting.
