<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] This is really the same as aid78's method.
$$(3+4i)e^{ix}=(3\cos x-4\sin x)+i(3\sin x+4\cos x)
=2+iA$$
for $A=3\sin x+4\cos x$. As $|(3+4i)e^{ix}|^2=3^2+4^2=25$
then $25=4+A^2$, so $A=\pm\sqrt{21}$ (both are possible).
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