<s>[INST] How to solve $\sin(x) + 2\sqrt{2}\cos x =3$ How to solve $\sin(x) + 2\sqrt{2}\cos x=3$ ?
What is general method for doing these kind of questions?
Thanks
 [/INST] I would emphasize that
$$ \sin (x + A) = \sin x \cos A + \cos x \sin A  $$
You have
$$ \sin x \; \; \frac{1}{3} + \cos x \; \;  \frac{\sqrt 8}{3} = 1   $$
We can take $A = \arctan \sqrt 8,$ so that $\cos A = 1/3$ and $\sin A = \sqrt 8 /  3. $ So, once again, you have
$$ \sin (x + \arctan \sqrt 8) = 1.  $$
On e of the answers is
$$ x = \frac{\pi}{2} - \arctan \sqrt 8 $$
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