<s>[INST] Given $3\cos x - 4 \sin x = 2$, find $3 \sin x + 4 \cos x$ without first solving for $x$ If $$3\cos{x}-4\sin{x}=2$$
find $$3\sin{x} +4\cos{x} $$
I have solved the equation for $x$, then calculated the required value, but I think there is a direct solution without solving the equation.
 [/INST] $$a \cos(x) - b \sin(x) = \sqrt{a^2+b^2} \cos(x+y)$$ where $\cos(y) = a/\sqrt{a^2+b^2}$ and $\sin(y) = b/\sqrt{a^2 + b^2}$.  Then $$a \sin(x) + b \cos(x) = \sqrt{a^2 + b^2} \sin(x+y)$$
In your case, with $a=3$ and $b=4$, $\sqrt{a^2+b^2}=5$, $\cos(x+y) = 2/5$ so $\sin(x+y) = \pm \sqrt{1-(2/5)^2} = \pm  \sqrt{21}/5$, so the answer is $\pm \sqrt{21}$. 
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