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Math Help - Simplification of Trig FUnctions

  1. #1
    Member classicstrings's Avatar
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    Simplification of Trig FUnctions

    How would I go about this one.?

    Simplify

    1/(sin@ + cos@)
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  2. #2
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    There's a standard technique for dealing with combinations such as sin(X) + cos(X), or, more generally, A.sin(X) + B.cos(X) for constants A and B (you have A=B=1). The idea is to reverse engineer the simplification sin(X+Y) = sin(X).cos(Y) + cos(X).sin(Y). So your first shot would be that you want A = cos(Y) and B = sin(Y), but that doesn't always happen, since sin^2(Y)+cos^2(Y) = 1, but it might not happen that A^2+B^2 = 1. So take out a factor of C = sqrt(A^2+B^2), and then look for a Y such that C.sin(Y) = A and C.cos(Y) = B. Dividing, tan(Y) = A/B, so that Y is arctan(A/B).

    In your case, A=B=1 so Y = arctan(1/1) = pi/4 and C = sqrt(1^2+1^2) = sqrt(2). Thus sin(X) + cos(X) = sqrt(2).sin(X+pi/4).
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  3. #3
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    Quote Originally Posted by rgep
    There's a standard technique for dealing with combinations such as sin(X) + cos(X), or, more generally, A.sin(X) + B.cos(X) for constants A and B (you have A=B=1). The idea is to reverse engineer the simplification sin(X+Y) = sin(X).cos(Y) + cos(X).sin(Y). So your first shot would be that you want A = cos(Y) and B = sin(Y), but that doesn't always happen, since sin^2(Y)+cos^2(Y) = 1, but it might not happen that A^2+B^2 = 1. So take out a factor of C = sqrt(A^2+B^2), and then look for a Y such that C.sin(Y) = A and C.cos(Y) = B. Dividing, tan(Y) = A/B, so that Y is arctan(A/B).

    In your case, A=B=1 so Y = arctan(1/1) = pi/4 and C = sqrt(1^2+1^2) = sqrt(2). Thus sin(X) + cos(X) = sqrt(2).sin(X+pi/4).
    rgep, nice explanation!

    This kind of question is very commonly posed on math help sites. I'd say I see it once or twice a week. Everyone who answers it must derive the solution, either from the sum formula as you have done or other ways. I'm looking for a good link to refer people to that contains your derivation.

    For example, this thread had this question in another guise. I found a link with the following explanation. But this explanation is not as complete as yours and the explanation is buried in the middle of a long page. Can you (or anyone else) give me more information on this technique (say names for it) to help me find a better link? Thank you.



    I found these formulas in Wikipedia, but there is no derivation.

    Sine formula.





    Cosine formula.

    Last edited by JakeD; September 7th 2006 at 03:26 PM.
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  4. #4
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    Quote Originally Posted by JakeD

    but there is no derivation.
    I am not satisfied with CaptainBlank's link.
    ---
    You have,
    A\sin x+B\cos x
    Write as, ( A,B\not = 0)
    \sqrt{A^2+B^2}\left( \frac{A}{\sqrt{A^2+B^2}}\sin x+\frac{B}{\sqrt{A^2+B^2}}\cos x\right)
    Let,
    \phi=\tan^{-1} B/A
    Then,
    \cos \phi=\frac{A}{\sqrt{A^2+B^2}}
    And,
    \sin \phi=\frac{B}{\sqrt{A^2+B^2}}
    Thus,
    \frac{1}{\sqrt{A^2+B^2}}(\cos \phi\sin x+\sin x \cos \phi)
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  5. #5
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    Quote Originally Posted by ThePerfectHacker
    I am not satisfied with CaptainBlank's link.
    ---
    1. It's not my link it's JakeD's, I was only pointing out that it had information
    needed to answer the original question in the thread it was posted in;
    because every one was ignoring it and in consequence going off at half-cock.

    2. I posted the image file because the needed information from the link was
    a long way down the page and might easily be missed.

    3. It was posted to answer the question: "what is the auxiliary angle method?".

    RonL
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