Biting and other hurtful behavior

What do you do for big behavioral infractions of a three year old,, like biting her younger sibling. Or hitting? She can get extremely aggressive, repeatedly.
Also, what should you do when they misbehave in public? Like screaming, ie “get your hands off me”?
This child is so strong willed that yesterday she had a screaming tantrum in her car seat for 40 minutes as we drove home. It was past her bedtime,so naturally she would be cranky, but 40 minutes if a tantrum wanting me to stop the car and let her out of her car seat to find a toy in the car. We told her she was in a calm down spot and that we wouldn’t talk to her until she got calm… but we did talk to her some over the 40 minutes. Reminding her to get calm, and at last to break her out of the tantrum, I had her repeat how a request should sound, Daddy, please, may I have…” I don’t know why that worked, but she calmed down after that, and we talked and gave an extra chore. What can or should be done to shorten tantrums. That was a very long 40 minutes.