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Question Number 72291 by aliesam last updated on 27/Oct/19

Answered by mr W last updated on 27/Oct/19

$${\left(5m\sin \left(3x\right)\right)}^2={\left(4\sin x\right)}^2$$$$5m\sin \left(3x\right)=\pm 4\sin x$$$$5m\sin x(3-4{\sin }^{2}x)=\pm 4\sin x$$$$\Rightarrow \sin x=0\Rightarrow x=k\pi$$$$\Rightarrow 5m(3-4{\sin }^{2}x)=\pm 4$$$$\Rightarrow {\sin }^{2}x=\frac{1}{4}(3\pm \frac{4}{5m})$$$$\Rightarrow \sin x=\pm \frac{1}{2}\sqrt{3\pm \frac{4}{5m}}$$$$\Rightarrow x=k\pi \pm {\sin }^{-1}\frac{1}{2}\sqrt{3\pm \frac{4}{5m}}$$$$with0⩽3\pm \frac{4}{5m}⩽4$$$$\Rightarrow m⩾\frac{4}{5}$$$$\Rightarrow m⩾\frac{4}{15}$$

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