That's a difficult question. You can't control your father emotion and neither can he. The only thing your father can control is his reaction to those emotions.
You think your father "get angry over nothing" but, for him, it certainly isn't nothing. The fact is, you don't know why he gets angry and, without the why, it will be hard to resolve the issue.
You said that you tried to ask him "why", but it failed. Based on personal experience, I think it was a good idea to try to understand the why. When you are trying to negotiate something, knowing why the other person doesn't want something is always a good idea.
But it failed, so what can you do now?
I think knowing the why is still your best shot but it won't come out easily.
As DDD said, timing is important. Don't ask when he is already preoccupied or angry, this will likely cause him to refuse the conversation and not answer you.
As your father might be embarrassed by the reason he gets angry, it's better to ask him privately. This way, he doesn't need to disclose the reason in front of everyone but only in front of you.
Edit: As WendyG pointed out (and I agree), when talking to people who are bad at "opening up", it's better to do it when you aren't face to face (like when you are on a walk or driving somewhere). I don't know why it works but, as someone who doesn't open up easily, I know that I more confortable this way.
Now, all this is good but might not be enough. In order to improve your chance of success, I suggest coming in with hypotheses of why your father gets angry. Warning: don't use them at the beginning the conversation, keep them for later if your father doesn't "open up".
Start the conversation by telling your father something like:
I would like to know why you don't want a male teacher to teach me. I feel that this might help me understand you better and I would like that.
As a person, I feel that getting to know someone better is a noble goal and your father might feel the same way. If he does, this will make him more likely to answer your request.
Give some time to your father to respond. If he doesn't, you might start throwing hypotheses:
Is it because X?
(for example: Is it because you are afraid the teacher might be a bad man?)
Let him answer, then give him the time to give an answer to the why question.
Then start the process again by throwing another hypothesis.
At any point, your father might close the discussion (I don't want to speak about it!). I won't tell you what to do in this case (push more or let it rest) but I would like you to keep this in mind:
Your father might not be ready to talk about his feeling, not yet and maybe not ever. Not with you and maybe not with anyone. He has the right to some privacy. If you want to respect his choice, you can tell him this:
I see that you don't want to talk about it. If you change your mind in the future, I will be there to listen to you.
The last sentence keeps the door open for future discussion, so that one day, you finally get the answer to why he his angry (and so that you can talk to him and peace his mind).
I feel that I have been a bit quick about why it's important to know the why. So, here it is:
Once you have the why, you can start negotiating. Without the why, you are just throwing random argument in the air, hoping that one will change your father mind. But, with the why, you can focus on what your father dislike and concentrate your effort on that. Which is, in my experience, way more efficient.