Knowing how to deal with a friend who is angry with you is never easy. You desperately want a resolution, but you don't feel you can communicate with your friend because of her anger. It is important to understand that your friend likely wants a resolution as well, but she wants you to initiate it. There are a few important factors to consider when it comes to winning back your friend's trust.
Approach him in private. If you want to show your friend that you respect him and sincerely want to resolve your conflict, approach him in a place where the two of you will not be interrupted and can have the needed time to discuss the issues you are having. Make sure to also speak to him at a time when he is not distracted by other things or attempting to focus on something else. Simply choosing the right time and the right place for conflict resolution will put him in the right frame of mind for what you are about to say.
Validate her feelings. Start off with validating the way she is feeling right now by explaining that you completely understand her anger. Instead of choosing to be defensive and asking her why she is angry at you, be understanding and point out that you also would be angry in her situation. By validating the way she feels, you are showing her that you are approaching the situation with empathy, instead of trying to prove that you are right and she is wrong.
Apologize sincerely. It may seem like a basic step, but an apology is the most important way to resolve conflict with your friend. A simple "I'm sorry" lets him know that you are dropping your pride in order to restore your damaged relationship and that you would rather be close to him than hold onto your ego. He will notice that you are being vulnerable and will likely open up to you and what you are suggesting.
Look for resolution. She may not be ready to speak to you just yet, and that is OK. Let her know that she can take the time she needs to move past the mistake you made and that you are patiently waiting for her to accept your apology. If she wants to spend time with you right away, consider yourself fortunate to have a friend who so willingly embraces your friendship.