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Trigonometric equation sin(x) = C*sin(y)

  1. Apr 10, 2016 #1
    1. The problem statement, all variables and given/known data
    sin x = C*sin y
    Find y as a function of x for a given C>0.

    2. Relevant equations
    sin x = C*sin y

    3. The attempt at a solution
    This is not actually a problem from a book, but a problem I myself thought about. I was studying elastic collisions in SCM and I obtained 2 equations:

    sin (Φ+α) = C*sin α
    sin (Φ-β)= sin β

    where C is the ratio of masses (C>0) and Φ, α, β∈[0,π].
    The second equation gives Φ∈{ 2β, π}.
    The solution Φ = π is false because when you substitute it in the first equation you get C=-1, contradiction with C>0.
    Now I want to solve the first equation to get Φ=f(α)=2β, but I have no idea how to solve that. If the equations are correct, then α+β=π/2 (known physics fact that can be proved in other ways; a very important result for billiard players).
    The equations may not be physically correct, but the mathematical equation sin x = C*sin y seems solvable and I have no idea how to solve it.
    I don't really want the solution, but a hint in solving it, or at least a starting point.
     
  2. jcsd
  3. Apr 10, 2016 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Take a look at the inverse trigonometric functions.
     
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