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Math Help - Proof of Compound Angles?

  1. #1
    Junior Member qspeechc's Avatar
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    Proof of Compound Angles?

    Could someone please give me the proofs, or point me to a website, of the compound angles formulae?
    That is:
    cos(A+B) = cosA.cosB - sinA.sinB

    And the other ones, cos(A-B), sin(A+B), sin(A-B), tan(A+B), tan(A-B).

    Are they just geometrical proofs, or are there algebraic ones? Thank-you.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by qspeechc View Post
    Could someone please give me the proofs, or point me to a website, of the compound angles formulae?
    That is:
    cos(A+B) = cosA.cosB - sinA.sinB

    And the other ones, cos(A-B), sin(A+B), sin(A-B), tan(A+B), tan(A-B).

    Are they just geometrical proofs, or are there algebraic ones? Thank-you.
    The only proofs I have seen are geometric.

    -Dan
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    The formal mathematical proof uses power series and the Cauchy product formula. But I doubt that is how you want it done.
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  4. #4
    Junior Member qspeechc's Avatar
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    Thank-you for the link topsquark, and TPH, I would like to see the power series proof, and you are correct in assuming that I do not know about Cauchy product formulae.
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  5. #5
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    Quote Originally Posted by qspeechc View Post
    Thank-you for the link topsquark, and TPH, I would like to see the power series proof, and you are correct in assuming that I do not know about Cauchy product formulae.
    The Power series proof is to know that,
    \sin x = \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{(2n+1)!}
    \cos x = \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n}}{(2n)!}
    You can to show,
    \sin(x+y) = \sin x \cos y + \cos x \sin y
    Now substitute those formulas and use the Cauchy product:
    \sum_{n=0}^{\infty}a_n \cdot \sum_{n=0}^{\infty}b_n = \sum_{n=0}^{\infty} c_n \mbox{ where }c_n = a_nb_0+a_{n-1}b_1+...+a_1b_{n-1}+a_0b_n.
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