<s>[INST] Determine sinus and cosinus : $2\sin x + 3\cos x = 3$ $$2\sin x + 3\cos x = 3$$
Seemed easy at first but I have no idea how to determine them. I tried replacing the 3 with $3(\sin^2 x + \cos x^2x)$ But it doesnt work. I also tried switching sides from almost everything and no luck. Please help!
 [/INST] Hint:
You can rewrite the equation as
\begin{align}
2\sin x +3\cos x=3&\iff 2\sin x=3(1-\cos x)\iff 4\sin\dfrac x2\cos\dfrac x2=6\sin^2\dfrac x2\\[1ex]
&\iff\begin{cases}
\sin \dfrac x2=0\\\;\text{ or}\\[-2ex]2\cos\dfrac x2=3\sin \dfrac x2
\end{cases}\iff\begin{cases}
\sin \dfrac x2=0\\\;\text{ or}\\[-2ex]\tan\dfrac x2=\dfrac23
\end{cases}\\&\iff\begin{cases}
\dfrac x2\equiv 0 \pmod\pi\\\;\text{ or}\\[-2ex]\dfrac x2\equiv\arctan\dfrac23\pmod{\pi}
\end{cases}\iff\begin{cases}x\equiv 0 \pmod{2\pi}\\\;\text{ or}\\x\equiv 2\arctan\dfrac23\pmod{2\pi}\end{cases}
\end{align}
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