Let's keep this simple:
This is already more than you need to know about the mechanics of breathing to do it well.
Most students DO NOT take deep-enough, relaxed-enough breaths when playing.
This is a new tip which I learned from Joe Alessi at the Alessi Seminar V in 2007. He would describe the ideal breath as being just like the deep breath you take at the doctor's office when he/she is listening with a stethoscope.
As you breathe in, hold one finger vertically in front of your mouth (as if you are saying, "shhh"). As you breathe deeply, you should hear a slight whistling sound as the air rushes around your finger. This has become my favorite breathing exercises because it is simple and it focuses the attention away from internal mechanics onto an external goal.
Say "keee." Your tongue is arched in the roof of your mouth. Now try to breathe deeply while holding the "keee" syllable. Terrible isn't it? Now say "hoe" (or "how"). Your tongue is low in your mouth allowing the air to flow smoothly. Breathe deeply with the "hoe" syllable. Did you get a better breath this time?
Simply cut the bottom off of a water or soda bottle (the opening
is sized for the mouth). Wash it if necessary.
Take a few deep breaths first without then with the tube in mouth. Did you
take in more air?
Now.... learn to take equally deep breaths without the tube and without
opening your mouth too wide.
You may want to make a breathing bag out of a lightweight paper bag. Insert the tube in the bag and completely fill and collapse it with each breath. Make the bag big enough so you can just barely fill it with your deepest breath. You may wish to purchase a medical breathing bag (somewhere between four and six liters). Plastic shopping bags are often too flimsy and tend to collapse around the end of the tube.
When playing your music, try pausing at each breath mark. Take as long as you need to complete a relaxed, dark "oh" breath before proceeding. After learning to consistently take this wonderful breath while pausing the music, gradually shrink the pause needed for the breath until you are playing the music in time while maintaining optimal breathing.
TIMING THE BREATH
Once you have developed the habit of taking a full relaxed breath, you must learn to do so in time with a metronome. All too often, young trombonists seem to abandon good breathing habits as soon as they are required to coordinate their breath with a steady pulse. Two common mistakes arise:
Keep the metronome turned on during your breathing exercises and warmup routine to help alleviate this problem.