Help finding maximum amplitude of sine function so that it fits inside a circle
Honestly just thinking about this and not sure how to solve for the non-trivial solution. Drew this on desmos with an approximate solution for demonstration purposes. Circle has radius π and the sine wave is approximately y = 2.64sin(x).
I first thought to do something algebraic like Csin(x) = sqrt(pi²-x²) and solving for C but I couldn't find a solution. I then tried to convert it to polar equations to simplify the side representing the circle, but I don't think theres a way to properly represent the cartesian sine function in a polar system. I also tried to use parametric equations and found some interesting manipulations involving nested inverse trig functions, but it didn't really get me anywhere.
Ultimately, through all this algebra, I realized I might be approaching the question wrong. I realized it's not really in the world of elementary algebra anyways since I'm doing more than solving for C. I'm looking for a value of C such that there are exactly four solutions (or two solutions if you just look at the right half of the circle).
Not super sure where to go from here. Would love to find the solution on my own, but I'm also just doing this out of curiosity, so I would also appreciate just the solution haha.