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Math Help - Find FI in: A = B.sin(FI) + C.cos(FI)

  1. #1
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    Jun 2009
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    Find FI in: A = B.sin(FI) + C.cos(FI)

    Hello all,

    I need to solve this problem. Can anyone give me some directions to follow or some reference to some article? Thanks in advance.

    The problem is: I need to find the angle FI in the following equation:

    A = B * sin(FI) + C * cos(FI)

    Best regards to all,
    Emmanuel.
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Dec 2008
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    South Coast of England
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    Hello Emmanuel

    Welcome to Math Help Forum!
    Quote Originally Posted by Emmanuel View Post
    Hello all,

    I need to solve this problem. Can anyone give me some directions to follow or some reference to some article? Thanks in advance.

    The problem is: I need to find the angle FI in the following equation:

    A = B * sin(FI) + C * cos(FI)

    Best regards to all,
    Emmanuel.
    Let b\sin\phi + c\cos\phi = r \sin(\phi+\alpha) = r\sin\phi\cos\alpha+r\cos\phi\sin\alpha

    Compare coefficients:

    r\cos\alpha = b (1)

    r\sin\alpha = c (2)

    Square and add:

    r^2(\cos^2\alpha+\sin^2\alpha) = b^2+c^2

    \Rightarrow r = \sqrt{b^2 + c^2}. This gives the value of r.

    Then divide (2) by (1) to get \tan\alpha = \tfrac{c}{b}, which gives the value of \alpha.

    Finally, r\sin(\phi+\alpha) = a \Rightarrow \phi = \arcsin(\tfrac{a}{r})-\alpha

    Grandad
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