solve for x: $\sin(ax)=k\sin(bx)$, a,b,k and x are real numbers
I am looking for a very general solution when a,b and k are completely unrelated.
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What if a,b,k are rational numbers? – user130354 Feb 27 '14 at 15:45
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No. There isn't one in general. Not unless you consider the following infinitely-nested expression acceptable: $$x=\frac1a\arcsin\left(k\sin\left(\frac ba\arcsin\left(k\sin\left(\frac ba\arcsin\left(k\sin\left(\frac ba\arcsin\left(k\sin\left(\ldots\right)\right)\right)\right)\right)\right)\right)\right)$$
Lucian
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That's a cute expression you have there. If you could provide a hint as to one goes about obtaining it, I would be most grateful. :) – David H Feb 20 '14 at 23:31
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2@DavidH: Just apply $\arcsin$ to both sides of the original equation, divide through a, and then replace x recursively. – Lucian Feb 20 '14 at 23:37
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