1. "Knowing what topics will be on the quiz is half the battle"
Start by asking the teacher tons of questions like "will we need to know this for the quiz?" or "is this one of the key problems that we should know how to solve?" or "would you say that this is a topic of major importance for us to learn in this class?"
If you can, look at the teacher's past quizzes and talk to former students (seniors) about this teacher to see what his tests are usually like. Do they look the same from year to year? Google terms like "inverse trig quizzes" to test yourself and compare what you find to what the teacher gives you.
2. "Be prepared"
Get enough sleep.
Eat a good breakfast. Use the bathroom before the quiz. Have extra paper and pencils. Bring your calculator with extra batteries.
Bring your "Note Sheet" with everything you need on it. Do NOT lose this. Don't put too much or too little information on it, and keep it organized and clear with topic headings based on which subjects you're going to need to know. Put everything on here that you don't want to have to memorize, if you're allowed. You want to know where to find everything you need on it and label it clearly, because you're going to be under time pressure and you don't want to stress yourself out by looking for things that are hard to find. The calmer and more organized you are, the better. Good test taking strategy, timing, and preparation is almost as important as studying.
3. "Recognize the question types and know what type of answers they're asking you for."
If there are five different major problem types that you expect to see in your quiz, study them until you can recognize what to do before you even write the problem down. Then do a practice quiz of ten questions, for example two questions from each of the five types. Time yourself. Keep repeating until you can finish the quiz in a time that you'd comfortable with on the quiz.
I'd expect your quiz to be broken up into at least three problem types: calculations (numbers), trivia facts (words, terms, and definitions), and graphs (arcsin, arccos, and arctan). Review these with examples from the book, and if you can, write down notes and tips on your Note Sheet.
5. "Remember the Problem Solving Procedure"
I learned this from Physics:
If a picture can be drawn to describe your problem, draw it and label it. Write down your givens. Write down your unknowns
As you begin to solve the problem, check off the unknowns (and give their values) until you've found what the question is asking you for and answered the question. ALWAYS check your work if you have the time.
6. "Timing is Everything"
If you've ever taken the SAT, you probably know that it's not only about how smart you are. It's about how fast you are. Who cares if you can get all the questions right but it takes you hours? I'd advise you NOT to solve the quiz questions in order. Start with the easy questions first, and then move on to the harder questions last. If you don't immediately recognize how to solve a question when you see it, skip it and go back to it later. Don't spend too much time on any one question. If you had 10 questions on a 60 minute quiz, no question should be taking up more than 5-6 minutes because you want to leave yourself some time at the end to check your answers.