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B What are some strategies to prove trigonometric results

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  1. May 11, 2017 #1
    I am facing a lot of problem in proving trigonometric results (advanced ones). There are a lot of formulas (compound angle related ones, transformation of sum into product ones and vice versa, multiple angle and sub-multiple angle ones). I am unable to figure out how to proceed forward in proving results and what formula to apply. I got 0 out of 20 in my exam on trigonometry.

    I would be very grateful if someone could provide some suggestions (which are helpful and applicable in almost all situations) on how to decide what formula to apply and how to proceed in general. I am sorry if my question appears to be too broad. I have not asked about any specific problem because that would be of little help to me in other problems.
     
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  3. May 11, 2017 #2

    BvU

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    Hard to judge this way on what level you want to be assisted. Are you familiar with complex numbers? Euler formula ? Fluent with all/most/half/some/hardly any of these identities ? Sine rule ? cosine rule ?
     
  4. May 11, 2017 #3
    As @BvU has said, more information would be helpful to those trying to help you. In addition to the question he asked you about your knowledge of specific identities etc., when you say this -
    - I don't necessarily agree; I think providing an example of a proof that stumped you, with sufficient background - e.g. how you tried to approach it and where you got stuck - might demonstrate how you approach problems in general, which could be useful.

    Along those lines, I noticed that your profile identifies you as a high school student; are you still in high school, or have you started freshman year in college? In either case, that connects to something else you said that stuck out for me:
    Why I think this matters: Personally, I'm very much a beginner in math, certainly not as far along as you are (see my profile for more information). However back in January I took a very useful course in predicate logic and elementary proofs, a MOOC created by Stanford math prof Keith Devlin, called "Introduction to Mathematical Thinking." If you haven't yet taken an introductory course on logic & proofs, I'd highly recommend this one; it's free & you can go at your own pace, very good instructional videos & assignments. Anyway, a big point Devlin makes throughout the course is that although proofs very much require knowing identities and definitions etc., the way you go about devising a proof typically isn't by starting with "hmmm what formula should I use here?" and just trying to crank, the way you would crank a calculation in high school. He wrote an inexpensive little book to go with the course; the following excerpt is from the introduction. I've edited it down a tiny bit, but it's still rather long; so I'm just going to indent it rather than quote it:

    . . . in the case of math, your goal at college is to develop the thinking skills that will allow you to solve novel problems (either practical, real-world problems or ones that arise in math or science) for which you don’t know a standard procedure. To put it another way, before college you succeed in math by learning to “think inside the box”; at college, success in math comes from learning to “think outside the box."

    The first key step (there are two) you have to make in order to successfully negotiate this school-to-college transition, is to learn to stop looking for a formula to apply or a procedure to follow. Approaching a new problem by looking for a template— say a worked example in a textbook or presented on a YouTube video — and then just changing the numbers, usually won’t work. (Working that way is still useful in many parts of college math, and for real-world applications, so all that work you did at high school won’t go to waste. But it isn’t enough for the new kind of “mathematical thinking” you’ll be required to do in many of your college math courses.)

    So, if you can’t solve a problem by looking for a template to follow, a formula to plug some numbers into, or a procedure to apply, what do you do? The answer— and this is the second key step— is you think about the problem. Not the form it has (which is probably what you were taught to do at school, and which served you well there), but what it actually says. That sounds as though it ought to be easy, but most of us initially find it extremely hard and very frustrating.​

    He elaborates throughout the course on what he means by "thinking" about the problem. Some of it is learning enough predicate logic to begin to make use of "if then" and similar statements; also some set theory which can be helpful. Another aspect is learning to brainstorm to find an intuitive understanding of a problem via a quick sketch or some other back-of-the-napkin technique; or via pondering "where am I starting from, and where do I want to wind up?" And another is learning various common strategies for proofs - e.g. contradiction, mathematical induction, etc. In his course you get a taste for all of these. Also useful for anyone who hasn't done a lot of proofs before is Richard Hammack's free PDF book The Book of Proof, available at this link; he walks you through many of the same points as Devlin, but in a bit more comprehensive manner. I checked and did find a trig problem or two in Book of Proof; they can be found in the chapter on functions.

    Lara Alcock in her book How To Study As A Mathematics Major also provides some helpful hints on moving from high school-level math to college level ditto; and on learning to do proofs by understanding why definitions exist & how to use these as building blocks in proofs. You might like her book if indeed you are making that transition or about to.

    (I apologize if all this about proofs isn't really relevant to your needs; when I saw that word I zeroed in on it.)
     
    Last edited: May 11, 2017
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