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

1. Apr 10, 2016

### anachin6000

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. Apr 10, 2016

### Samy_A

Take a look at the inverse trigonometric functions.