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Sum of sine and cosine

  1. Sep 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Show that any sum:
    Asin(α) + Bcos(α)

    can be written as : C sin(α+ϕ)
    2. Relevant equations



    3. The attempt at a solution
    i can express cos(a) as as sin(90-a), and then try to use the formula that adds sines, but it gives the form of cos*sin.



     
  2. jcsd
  3. Sep 22, 2016 #2
    After expressing C sin(α+ϕ), i managed to get the following:
    A = C*cos(phi), B = C*sin(phi),

    C = (A+B)/ (Cos(phi) + Sin(phi))
     
    Last edited: Sep 22, 2016
  4. Sep 22, 2016 #3

    SammyS

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    One backwards parenthesis .

    Once you get A = C⋅cos(ϕ) and B = C⋅sin(ϕ),

    Square each equation, then add them.
     
    Last edited: Sep 22, 2016
  5. Sep 22, 2016 #4
    A^2+B^2 = C2cos2phi + C2sin2phi = C2
    C = sqrt(A2+B2)
     
  6. Sep 22, 2016 #5
    then i can devide B/A = tan (Phi)
    and then arctan B/A = Phi

    Then we got Csin(α + arctan(B/A)).

    Thanks
     
  7. Sep 22, 2016 #6

    SammyS

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    You may need to be careful about the signs of A and B. (Well in this case, just the sign of A.)

    If A is negative, then so is cos(ϕ), and that puts ϕ in the 2nd or 3rd quadrant.
     
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