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Question Number 146310 by mathdanisur last updated on 12/Jul/21

$$4sin\left(x\right)cos\left(x\right)⩾\sqrt{3}$$

Commented by mathdanisur last updated on 12/Jul/21

$$Solvethetrigonometricinequality$$

Commented by MJS_new last updated on 12/Jul/21

$$\mathrm{let}{a}_k=\frac{\pi }{6}+k\pi \wedge b_k=\frac{\pi }{3}+k\pi \wedge k\in \mathbb{Z}$$$$\Rightarrow$$$$4\sin x\cos x⩾\sqrt{3}\iff a_k⩽x⩽b_k\mathrm{\forall }k\in \mathbb{Z}$$

Commented by mathdanisur last updated on 13/Jul/21

$$thanksSer$$

Answered by puissant last updated on 12/Jul/21

$$4\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)=2\sin \left(2\mathrm{x}\right)$$$$\Rightarrow 2\sin \left(2\mathrm{x}\right)⩾\sqrt{3}$$$$\Rightarrow \sin \left(2\mathrm{x}\right)⩾\frac{\sqrt{3}}{2}=\sin \frac{\pi }{3}..$$

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