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Trigonometric manipulation

  1. Nov 1, 2013 #1
    Hello,

    Probably a simple problem, but Im not able to figure it out.

    [tex] a \cos (\epsilon) - b \sin (\epsilon) = c [/tex] in-phase part
    [tex] a \sin (\epsilon) - b \cos (\epsilon) = d [/tex] out-of-phase part

    In order to find the phase shift, the in-phase term has to be divided by the out-of-phase term?

    [tex] \frac{a \cos (\epsilon) - b \sin (\epsilon) = c}{a \sin (\epsilon) - b \cos (\epsilon) = d} [/tex]

    And the phase shift is the arctan of the out of phase and the in-phase term to my knowledge. But I am not able to manipulate the formula in a way that I'll get to an arctan. Can someone point me in the right direction?

    Thanks
     
  2. jcsd
  3. Nov 1, 2013 #2
    $$ a \cos x + b \sin x = \sqrt {a^2 + b^2 } ( \frac a {\sqrt {a^2 + b^2} } \cos x + b \frac b {\sqrt {a^2 + b^2} } \sin x)
    \\ = \sqrt {a^2 + b^2 } ( \sin c \cos x + \cos c \sin x) = \ ...
    $$
     
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