The number of values of x for which sin 2x + | vedclass

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(A) We are given the equation $\sin 2x + \sin 4x = 2$.Since the maximum value of the sine function is $1$,the equation $\sin 2x + \sin 4x = 2$ can onl

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(A) We are given the equation $\sin 2x + \sin 4x = 2$.Since the maximum value of the sine function is $1$,the equation $\sin 2x + \sin 4x = 2$ can only hold if $\sin 2x = 1$ and $\sin 4x = 1$ simultaneously.For $\sin 2x = 1$,we have $2x = 2n\pi + \frac{\pi}{2}$,which implies $x = n\pi + \frac{\pi}{4}$ for any integer $n$.For $\sin 4x = 1$,we have $4x = 2m\pi + \frac{\pi}{2}$,which implies $x = \frac{m\pi}{2} + \frac{\pi}{8}$ for any integer $m$.If we substitute $x = n\pi + \frac{\pi}{4}$ into $\sin 4x$,we get $\sin(4(n\pi + \frac{\pi}{4})) = \sin(4n\pi + \pi) = \sin(\pi) = 0$.Since $0 
eq 1$,there is no value of $x$ that satisfies both equations simultaneously.Therefore,the number of values of $x$ is $0$.

The number of values of $x$ for which $\sin 2x + \sin 4x = 2$ is

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