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I have been attempting to isolate $x$ in this equation and I thought I found a solution, but on graphing this, the output is undefined. I'm sure I've made an error in one of my steps but I have no clue about which one. $$ax=\sin(bx+c)$$

These are the steps I've used during my calculations. $a$, $b$ and $c$ are all positive real numbers. Any help would be greatly appreciated. $$ax=\sin(bx)\cos(c)+\cos(bx)\sin(c)$$ $$\frac{d}{dx}ax=\frac{d}{dx}\sin(bx)\cos(c)+\frac{d}{dx}\cos(bx)\sin(c)$$ $$a=b\cos(c)\cos(bx)-b\sin(c)\sin(bx)$$ $$\frac{a}{b}=\cos(c)\cos(bx)-\sin(c)\sin(bx)$$ $$\frac{a}{b}=\cos(c-bx)$$ $$\arccos(\frac{a}{b})=c-bx$$ $$bx=\arccos(\frac{a}{b})-c$$ $$x=\frac{\arccos(\frac{a}{b})-c}{b}$$

Schrödinger's Cat
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    You cannot differentiate both sides as there the solutions $x$ are not an interval, but rather many distinct numbers/a number. For example, $x-1=2$ has one distinct solution $x=3$, so differentiating both sides fails while $\sqrt{x^2}=x$ is true for $x$ on $[0,\infty)$, so both sides can be differentiated. You may use this post to solve your equation. – Тyma Gaidash Dec 15 '23 at 02:05
  • There is no closed form solution for $x$ in this equation. – Robert Israel Dec 15 '23 at 05:02

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