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Combining two trig terms into one?

  1. Jan 24, 2010 #1
    If I have 2 cosine terms added together, how would I combine them into one cosine term?

    Ex:
    A) 3 cos(2t)
    B) cos(2t - pi/2)

    Thanks

    PS. I don't think the sum to product formulas work, I'm wondering how to combine them into a single cosine term?
     
  2. jcsd
  3. Jan 24, 2010 #2
    These kind of operations are easiest with complex numbers.
    [tex]3\cos(2t)+\cos(2t-\frac{\pi}{2})=\Re\left(3\exp(2ti)+\exp(2ti-\pi i/2)\right)=\Re\left(\exp(2ti)(3-i)\right)[/tex]
    [tex]=\Re\left(\exp(2ti)\sqrt{10}\exp(-i\arctan\frac{1}{3})\right)=\sqrt{10}}\cos(2t-\arctan(\frac{1}{3}))[/tex]
     
  4. Jan 25, 2010 #3

    Mentallic

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    Homework Helper

    You can solve this by the use of the auxillary angle technique.

    [itex]cos\left((2t)-\pi/2\right)=sin(2t)[/itex] (you can confirm this by expanding the LHS, but this is a trigo identity you may remember having learnt).

    Now let [itex]3cos(2t)+sin(2t)\equiv Rsin(2t-\theta)[/itex]

    expand the RHS and then equate like terms. Solve the system of 2 equations in R and [itex]\theta[/itex] and then you'll have the original equation in terms of just one trigonometrical expression.
     
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