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Math Help - simplifying sin and cos

  1. #1
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    simplifying sin and cos

    i really need help in solving these, including steps.. help would be appreciated..

    show that.. sina/(1+cosa) + (1+cosa)/sina = 2/sina


    thankyou guys
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  2. #2
    Super Member angel.white's Avatar
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    start with:
    \frac{sin(a)}{1+cos(a)}+\frac{1+cos(a)}{sin(a)}

    Get common denominator:
    \frac{sin(a)}{sin(a)}*\frac{sin(a)}{(1+cos(a))}+\f  rac{(1+cos(a))}{sin(a)}*\frac{1+cos(a)}{1+cos(a)}

    Simplify:
    \frac{sin^{2}(a)}{sin(a)(1+cos(a))}+\frac{(1+cos(a  ))^{2}}{sin(a)(1+cos(a))}

    Use the equation that says sin^{2}(x) + cos^{2}(x)=1 to see that sin^{2}(x)=1-cos^{2}(x) and substitute that into your equation:

    \frac{1-cos^{2}(a)}{sin(a)(1+cos(a))}+\frac{(1+cos(a))^{2}  }{sin(a)(1+cos(a))}

    Combine the fractions:
    \frac{1-cos^{2}(a)+(1+cos(a))^{2}}{sin(a)(1+cos(a))}

    Multiply out the squared term:
    \frac{1-cos^{2}(a)+1+2cos(a)+cos^{2}(a)}{sin(a)(1+cos(a))}

    Add like terms:
    \frac{2+2cos(a)}{sin(a)(1+cos(a))}

    Factor out a 2:
    \frac{2(1+cos(a))}{sin(a)(1+cos(a))}

    1+cos(a) is in the numerator and the denominator so they cancel eachother out:
    \frac{2}{sin(a)}
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  3. #3
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    thankyou heaps, your amazing!
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  4. #4
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    angel.white... what beats me is how people like yourself know the exact steps to undertake to solve that problem the the exact simplified figure... coz there are 100's of ways to go about each step and thats the problem im having with these questions....
    so could you give me any advise?
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  5. #5
    Super Member angel.white's Avatar
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    Well, I didn't know the exact steps, but I did have a general idea where I was going (I worked it on paper until I could see that I definitely was giong to get it)

    For the general idea, there are usually intuitive next steps. When you have a fraction like you did, the intuitive next step is to get a common denominator and combine them. When you see sin^{2}(x) or cos^{2}(x) the equation sin^{2}(x)+cos^{2}(x)=1 immediately comes to mind (memorize this equation). When you have an equation like (a+b)^{2} the intuitive next step is a^{2}+2b+b^{2}, when you have like terms, the intuitive next step is to combine them, when you have clearly factorisable terms like (ab+ac) the intuitive next step is a(b+c). and when you have the same term multiplied by the numerator and the denominator, the intuitive step is to remove them.

    The next intuitive step won't always do you good, but it generally gives you a good direction to go in, and problems given courses like trig are usually constructed in such a way that the intuitive steps will come out correctly, and your answers will end up nice and neat (not so much for me in my calc class right now, though lol).

    So I guess there are 2 important things, the first is to be familiar with the equations, that way you see when you have opportunities to use them (earlier this semester, I made the mistake of not being familiar with the double angle formula, and spent 45 minutes trying to figure out this problem that took me 2 minutes to do once I realized that the double angle formula existed), and the second is to know where you are trying to go, to have that direction you are working towards so that the steps you follow make progress and lead you closer to your desired answer. This is important, because you can do and undo any step along the way, but you don't want to work backwards, so it's good to know which way you're going. (I did this too, on a test 2 weeks ago, worked this problem for a few minutes, and ended up back at the same equation I had started with lol).

    Anyway, good luck ^_^
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