The pipes in our master bathroom have just started making a banging sound after the toilet is done flushing and when the cold water is turned off. I tried turning off the water and draining the pipes but the banging continues. We didn't have any plumbing work done before the sound started.

One of the "Water hammer" happens when you abruptly shut off the water. You have at least 40 pounds of pressure pushing the water flow. The weight of the water hitting a 90 degree bend causes the sound that you hear. We put in air chambers to cushion the hammer much like the shocks on your car.
Back in the 60's we were required to to put air chambers on all bath lavatories and kitchen sinks. Over time water and condensate built up in them and they must be recharged. First turn off the house at the main house shut off valve. Then open hot and cold faucets at the farthest bathroom to prevent air lock.(This is important.) You will find under each fixture little brass or chrome valves. These are called angle stops. With a small pail held under the angle stop to catch the run off look under the handle of the stop and you will see a nut with the stem running through it. This is called the bonnet packing nut. Put your wrench on it and back it off counterclockwise. Now open the stop and remove the stem and washer assembly, The stop will began to drain. When it has quit draining reassemble the stop and move on to the next one. Do this on each one of your angle stops. There will be six of them in a two bath home. When you are done, close off the faucets you have opened, turn on the water to the house, and give yourself a pat on the back.
You have just recharged your air chambers and saved yourself a hefty service call bill. Hope this helps you out and thank you for rating my reply. TOM