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Trigonometry elementary problem

  1. Jul 9, 2012 #1
    1. The problem statement, all variables and given/known data
    If 3 sin theta + 5cos theta = 5 ... prove that 5sin theta - 3 cos theta = +- 3


    2. Relevant equations



    3. The attempt at a solution

    i tried many things here , mostly those including squares , because i need +-3 , but this one has me stumped :/
     
  2. jcsd
  3. Jul 9, 2012 #2

    Curious3141

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    Are you familiar with this "trick": [itex]a\sin \theta + b\cos \theta = \sqrt{a^2 + b^2}\sin(\theta + \arctan{\frac{b}{a}})[/itex]?
     
  4. Jul 9, 2012 #3
    never heard of it
     
  5. Jul 9, 2012 #4

    Curious3141

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    It's basically the angle sum formula for sine. Try expanding [itex]R\sin(\theta + \alpha)[/itex] and compare coefficients to [itex]a \sin \theta + b \cos \theta[/itex].
     
  6. Jul 9, 2012 #5
    can this question be solved without it ?>
     
  7. Jul 9, 2012 #6

    Curious3141

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    Not easily, I think. But why don't you want to try this? Haven't you covered the angle sum formula at all?
     
  8. Jul 9, 2012 #7
    You can try writing [itex]5cosθ=5\sqrt{1-sin^2θ}[/itex]. Taking this term to RHS,
    you will get an equation
    [tex]3sinθ-5=-5\sqrt{1-sin^2θ}[/tex]
    Square both the sides, the equation will be easy to solve and you will get two values for θ.
     
  9. Jul 9, 2012 #8

    SammyS

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    Very good ! :smile:
     
  10. Jul 9, 2012 #9
    thank you!!!! that did the trick
     
  11. Jul 9, 2012 #10
    Thank you SammyS! :smile:
     
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