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Summing sines and cosines with trig.

  1. Sep 28, 2013 #1
    Hey guys,

    I just wanted to come here and see if anyone could help me out. I am getting stuck using the trig formulas for summing sines and cosines. I won't beat around the bush too much, so let's get right into it.

    1. The problem statement, all variables and given/known data
    Use a trig formula to write the two terms as a single harmonic.
    sin(2x) + sin(2(x+(pi/3)))

    2. Relevant equations
    sin(A+B) = sinA*cosB + sinB*cosA
    cos(A+B) = cosA*cosB - sinA*sinB
    sin^2(A) + cos^2(A) = 1

    3. The attempt at a solution
    So I think that I might be over-thinking this, or maybe I am just not recognizing some pattern which would lead me to the solution. I'll walk through my steps:

    1. I assume that to solve this, I should expand these two functions with the trig formulas I mentioned above. I am hoping that in doing so, I can combine like terms and find a single sine or cosine function which gives me all the information I need.

    2. The given problem is equivalent to the expression: sin2x + sin(2x + 2pi/3)

    3. From that follows:
    2*sinx*cosx + sin2x*cos(2*pi/3) + cos2x*sin(2*pi/3)

    4. From that, I get:
    2*sinx*cosx - (1/2)sin2x - (sqrt(3)/2)cos2x

    5. Following that:
    2*sinx*cosx - sinx*cosx - (sqrt(3)/2)(cos^2(x) - sin^2(x))

    6. Then I get:
    sinx*cosx - (sqrt(3)/2)(cos^2(x) - sin^2(x))

    7. Which ultimately leads me to:
    (1/2)sin2x - (sqrt(3)/2)cos2x

    Step seven is where I get stuck. I honestly have no clue where to go from here. Any tips or hints would be greatly appreciated!!
     
  2. jcsd
  3. Sep 28, 2013 #2

    Dick

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  4. Sep 28, 2013 #3
    Thanks! Would it be 2(x + phi) or 2x + phi?
     
  5. Sep 29, 2013 #4

    Dick

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    2x+phi. Just substitute 2x for x into the formula. That doesn't change anything does it?
     
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