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Trig identities

Fizban140Fizban140 Registered User, __BANNED USERS regular
edited July 2011 in Help / Advice Forum
I have a trig final on tuesday and I can't figure out identities. Like figuring out what tanx/2 is equal to or any of it except cos A+/-B and sin A+\-B. I obviously know all the easy shit like sin^2+cos^2 = 1 and all of those but I just can't do these massive algebraic conversions. For example

I am trying to prove that (sin^2x)/2-2cosx = cos^2 x/2 and I get to a point where I have 1-cos^2\2(1-cosx) = cos^2 x/2 which I can cancel out to be 1+cosx/2 = cos^2 x/2

I don't get how those equal eachother, it doesn't make sense to me that a function squarerd and divided by two could equal a function plus 1 and divided by two.

Fizban140 on

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    KorlashKorlash Québécois TorontoRegistered User regular
    edited July 2011
    I don't know what you find to be so confusing. Why wouldn't you be able to do these operations to a function and arrive at an equal function? If you can show that the functions are equivalent through algebra, then you can only admit that they're equal.

    I'm not sure I totally follow the way you're writing these equations, but I understand that to complete your proof, you need to prove that [cos(2x) +1]/2 = (cos x)^2 ?

    Start with the identity for cos(A + B). What if A = B?

    The expression you will obtain will link cos2x to (cos x)^2 and (sin x)^2. From this, you can use (cos x)^2 + (sin x)^2 = 1 to convert your expression in terms of cosines.

    Korlash on
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    Fizban140Fizban140 Registered User, __BANNED USERS regular
    edited July 2011
    I don't understand how to find sinA\2
    I know that sin2a is equal to sin(A+A) so sinAcosA+cosAsinA = 2sinAcosA is that correct? So now to find sinA/2 what do I do? if I know that sin2a=2SinAcosA then where do I go? I have no idea how to even begin.

    Fizban140 on
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    Dunadan019Dunadan019 Registered User regular
    edited July 2011
    trig identity of cos(x/2):

    cos (x/2) = ((1+cos(x))/2)^.5

    so cos^2(x/2) = (1+cos(x))/2

    or were you looking for a proof?

    Dunadan019 on
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    Fizban140Fizban140 Registered User, __BANNED USERS regular
    edited July 2011
    I have the proof I just don't understand how I can do this in a year from now when I have forgotten all this shit. I am zero interest in memorizing forumlas, I want to learn math.

    Fizban140 on
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    Dunadan019Dunadan019 Registered User regular
    edited July 2011
    well, half of learning anything is remembering where to look up the material that you'll forget.

    I didn't happen to have the half angle formula for cosine floating around my head, I looked it up.

    is there some reason you'll have to do trig transformations a year from now?

    Dunadan019 on
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    Fizban140Fizban140 Registered User, __BANNED USERS regular
    edited July 2011
    Well I am taking calculus in 2 months or so so I am trying to really learn all of this trig really well before then. Also going for an engineering degree so I really want to learn math well, but I don't know how.

    Fizban140 on
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    ceresceres When the last moon is cast over the last star of morning And the future has past without even a last desperate warningRegistered User, Moderator mod
    edited July 2011
    Uh, learning trig identities at the algebra stage pretty much comes down to "See this list of shit? Memorize it, fucker." You don't have to understand it for like 30 more levels, and then you get a quest.

    Seriously. You don't have to know why right now. You just have to know that it works and remember it so you can use it. And depending on the kind and level of Calc you take, you won't even need to know it then.

    ceres on
    And it seems like all is dying, and would leave the world to mourn
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    Fizban140Fizban140 Registered User, __BANNED USERS regular
    edited July 2011
    I am just taking calc 1 next semester, but I am going for an engineering degree eventually.

    What do I need to learn for calculus? I plan on making use of my time off.

    Fizban140 on
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    ceresceres When the last moon is cast over the last star of morning And the future has past without even a last desperate warningRegistered User, Moderator mod
    edited July 2011
    You sound like you need to take pre-calc, honestly. It's a pretty good bridge between college trig and calc I. It ends up being geometry/algebra review, which it sounds like you desperately need. Some teachers will even touch briefly upon derivatives, and that is incredibly useful.

    But get used to just memorizing formulae - especially the trig stuff. For Calc I for majors (which you will probably have to take) you're going to have to do more than memorize a list of trig identities... you're going to have to memorize pages and pages of their derivatives, too. My teacher didn't make us memorize them, he just gave us pages and pages of formulae to use and said "know how to recognize and apply these". I know other teachers in the department wanted their students to remember them all, so I consider myself lucky.

    You are going to have to get used to memorizing shit you don't know what it means and will only learn later, MAYBE.

    ceres on
    And it seems like all is dying, and would leave the world to mourn
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    Fizban140Fizban140 Registered User, __BANNED USERS regular
    edited July 2011
    I already took college algebra, pre calc is algebra and trig together.

    Fizban140 on
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    Dunadan019Dunadan019 Registered User regular
    edited July 2011
    Go to your college library and take a look at old tests and or homework assignments for calc 1 classes. or look online if you have a student resource for it.

    most likely, they will not involve trig identities, just the regular sin, cos, tan, cot, sec, csc.

    when you learn the chain rule they might do trig identities but not in the way that you'll have to know what cos(x/2) is transformed into.

    Dunadan019 on
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    ceresceres When the last moon is cast over the last star of morning And the future has past without even a last desperate warningRegistered User, Moderator mod
    edited July 2011
    I didn't say college algebra. I said college trig. I'm assuming that's what you're taking now because you said you have a trig final. I am recommending you take pre-calculus as well before calc I for review and reinforcement, because pre-calculus is a good review for both algebra (which you will need for calculus) and geometry (which you will need for calculus). There will also be some trig (which you will need for calculus), and they may touch upon derivatives briefly (which you will need so much for calculus it will make you nauseous).

    ceres on
    And it seems like all is dying, and would leave the world to mourn
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    DemerdarDemerdar Registered User regular
    edited July 2011
    It's not all that bad though. You will always apply the stuff you memorize, and it will just re-enforce it in your head. Like, I'll never forget the derivative of ln(x) because I use and see it all the time. The derivative of arccos(x)? I have no fucking clue, so I look it up in a table.

    Trig identities too. I have forgotten all but a few (but they do come in handy), so when I need them I look them up. If you are in a trig class though, you absolutely need to memorize them, because that's the whole point of your class.

    All in all I echo cere's advice, take precalc before jumping into calculus. As an engineer I took precalc before calc and it was a wise choice. It gets you back into the swing of things, you learn fundamentally what a function is as well as a somewhat thorough review of trig, algebra 2 (solving systems of equations) and all that jazz.

    Get used to memorizing though. It's not all that bad, it's not like biology or something. The interesting thing about mathematics is that the shit you memorize and apply now can be used throughout your entire career (at least as an engineer).

    Demerdar on
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    Dunadan019Dunadan019 Registered User regular
    edited July 2011
    I'd imagine he's already picked his classes for next semester (which isn't such a hard thing to change if needed)

    but my personal experience is that precalc was a waste of time.

    I suggest trying out calc 1 and for a few weeks and talk to the professor when you start, if you're doing badly or aren't confident then I suggest dropping the class and trying to get into a precalc class.

    if you want to study up before hand, go with

    geometry: area, surface area and volume of basic shapes, graphing
    algebra: fractions, solving equations, basic trig functions, multivariable equations
    other areas to look into: sequences and series, limits

    Dunadan019 on
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    Fizban140Fizban140 Registered User, __BANNED USERS regular
    edited July 2011
    I was told not to take pre calc because it is literally my college algrebra and trig class combined, which I just took. I should know everything from those classes already. I mean I am trying to get through these classes fast, that is why I took trig during the summer. I am 24 already and just starting college, pretty far behind already.

    Fizban140 on
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    UsagiUsagi Nah Registered User regular
    edited July 2011
    Repetition builds confidence and familiarity, and if you're struggling with trig identities you're not helping yourself by rushing into Calc I.

    Also, as a general rule, taking math over the summer short sessions is a Bad Idea unless you're just brushing up - the classes are compressed into a much shorter time span and you definitely don't learn as much as you would during a full semester.

    As for the final, you might want to corner your TA/GSI and ask for some help, as that's one of the reasons that they're there. But yes, trig identities and geometry formulae are just there for you to memorize right now, you don't need to worry about the why until Calc II/III.

    Usagi on
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    Fizban140Fizban140 Registered User, __BANNED USERS regular
    edited July 2011
    No TA or GSI, I was told though that the trig class was not at all condensed and we covered the entire book. It was 10 hours a week in class.

    Fizban140 on
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    KorlashKorlash Québécois TorontoRegistered User regular
    edited July 2011
    Just to clear up one thing: doing proofs is not really about memorizing facts.

    Mathematicians can do many proofs with just a piece of paper and a pen, not because they've got thousands of proofs memorized in their heads, but because they have developed a way of thinking that allows them to figure out how to do those proofs.

    They start with what they know, and keeping the goal in mind, they work their way through a proof. They'll try a bunch of stuff that doesn't work, or make mistakes along the way, but they'll eventually find the right way.

    I didn't have to look up anything to make my post because I knew what I wanted to prove and had memorized a few basic trig facts (then again, it wasn't a very complicated proof, but the same principles apply). So you do need to memorize a few things, but it really is not as bad as you think. You are of course more efficient if you remember more things, but many things can be done with just a few hazy memories and something to write on.

    At any rate, as an engineer, you won't have to do super complex mathematical proofs as part of your job. Just make sure that you know where your mathematical knowledge comes from, so that you know what to use as a reference.

    Korlash on
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    big lbig l Registered User regular
    edited July 2011
    If you are going into engineering, why would you memorize trig identities unless a grade depended on it? We invented computers for a reason. I was a math major and did some very serious calculus and I don't think anybody ever asked me "Off the top of your head, what's cos(x^2) equal to?" Learn Matlab or R or something useful instead, that will actually help you solve problems you will actually face. If you want to learn more about doing proofs I recommend Intro to Mathematical Reasoning by Eccles.

    Edit: Also it's been years since I took Calc 1 but I don't recall much if any trig. Calculus is more about limits and series.

    big l on
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    RikushixRikushix VancouverRegistered User regular
    edited July 2011
    Yeah man. I took engineering calculus and they literally give you a page (or several) of identities and formulas. Some of them you need to memorize, some of them are provided for you on tests. No one really cares about the proofs - just know how they apply to what you're solving.

    Rikushix on
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    Fizban140Fizban140 Registered User, __BANNED USERS regular
    edited July 2011
    Well mostly I was on another forum asking some homework questions and they said I should know how to derive identites for everything (I can get most except the half angle, double angle stuff and I think I can do double angle now) and that it is an important math skill.

    But also I have a final on tuesday.

    Fizban140 on
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    ED!ED! Registered User regular
    edited July 2011
    Well, technically you CAN derive all trig identities from the sum of two angles formulas you should have learned (Sin(A+B), Cos(A+B)) - but no one really expects you to sit on an exam and do that; so yes there is an expectation of KNOWING what that Sin(2A) is 2Sin(A)Cos(A) right off the top of your head. If you happen to forget, well you can easily use the above formulas to derive it, but those are ones you should know pretty quickly.

    As for Pre-Cal, I suppose its good to see all the material again - but you are correct in thinking it is nothing but a review of the material you've covered already, and an introduction to material you will cover again. The insane thing about Pre-Cal is that the teacher will come at the material assuming you HAVENT seen it before. So is Pre-Cal an absolute pre-req for Calc 1 - no, especially if you are not required to take. Will it hurt - of course not. However if your math game is good and you can run through a Pre-Cal book (at least the first 2/3's) you should be ok.

    ED! on
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    ceresceres When the last moon is cast over the last star of morning And the future has past without even a last desperate warningRegistered User, Moderator mod
    edited July 2011
    Yeah, judging from this and other of Fizban's threads, Fizban's math game is not good. Which is why the recommendation for pre-calc.

    Not because it isn't repetition, but because it is very, very clear that you NEED the repetition in order to be successful with this. There's no shame in this.

    Maybe find out what textbook they use? Because my Calc I class was like 5 solid weeks of trig crap and its derivations. I had a pretty tough time with limits, too, but that they don't touch in precalc.

    ceres on
    And it seems like all is dying, and would leave the world to mourn
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    big lbig l Registered User regular
    edited July 2011
    ceres wrote: »
    Yeah, judging from this and other of Fizban's threads, Fizban's math game is not good. Which is why the recommendation for pre-calc.

    Not because it isn't repetition, but because it is very, very clear that you NEED the repetition in order to be successful with this. There's no shame in this.

    Maybe find out what textbook they use? Because my Calc I class was like 5 solid weeks of trig crap and its derivations. I had a pretty tough time with limits, too, but that they don't touch in precalc.

    Personally I was never able to learn shit just from a textbook, I needed someone to do it in front of me, but others may learn differently. I think there are two kinds of things you learn in math - things you learn because you need to know them to do later stuff, and things you learn because there is a test next week. I think trig identities are the latter, especially for an engineering student who will be spending a lot of time with a calculator in hand once they get in-program. I would much rather have a rigorous understanding of limits and series. I only really used trig identities when I got to advanced calc and you are expected to switch between cartesian and polar coordinates every 5 minutes.

    big l on
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    ceresceres When the last moon is cast over the last star of morning And the future has past without even a last desperate warningRegistered User, Moderator mod
    edited July 2011
    big l wrote: »
    ceres wrote: »
    Yeah, judging from this and other of Fizban's threads, Fizban's math game is not good. Which is why the recommendation for pre-calc.

    Not because it isn't repetition, but because it is very, very clear that you NEED the repetition in order to be successful with this. There's no shame in this.

    Maybe find out what textbook they use? Because my Calc I class was like 5 solid weeks of trig crap and its derivations. I had a pretty tough time with limits, too, but that they don't touch in precalc.

    Personally I was never able to learn shit just from a textbook, I needed someone to do it in front of me, but others may learn differently. I think there are two kinds of things you learn in math - things you learn because you need to know them to do later stuff, and things you learn because there is a test next week. I think trig identities are the latter, especially for an engineering student who will be spending a lot of time with a calculator in hand once they get in-program. I would much rather have a rigorous understanding of limits and series. I only really used trig identities when I got to advanced calc and you are expected to switch between cartesian and polar coordinates every 5 minutes.
    I understand what you are trying to say about the usefulness about these things for your future career in engineering. What I am saying is that he will have no future career in engineering if he doesn't pass calculus.

    ceres on
    And it seems like all is dying, and would leave the world to mourn
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    big lbig l Registered User regular
    edited July 2011
    And you are right. I guess maybe I had forgotten how much trig you had to know for calc 1.

    big l on
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