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Question Number 207588 by hardmath last updated on 19/May/24

$$\sqrt{2}\sin \mathbf{x}+\cos \mathbf{x}⩾1$$

Answered by mr W last updated on 19/May/24

$$\frac{\sqrt{2}}{\sqrt{3}}\sin x+\frac{1}{\sqrt{3}}\cos x⩾\frac{1}{\sqrt{3}}$$$$\sin \alpha \sin x+\cos \alpha \cos x⩾\frac{1}{\sqrt{3}}$$$$with\alpha ={\cos }^{-1}\frac{1}{\sqrt{3}}$$$$\cos (x-\alpha )⩾\frac{1}{\sqrt{3}}$$$$2k\pi -{\cos }^{-1}\frac{1}{\sqrt{3}}⩽x-\alpha ⩽2k\pi +{\cos }^{-1}\frac{1}{\sqrt{3}}$$$$\Rightarrow 2k\pi ⩽x⩽2k\pi +2{\cos }^{-1}\frac{1}{\sqrt{3}}$$$$or$$$$\Rightarrow 2k\pi ⩽x⩽(2k+1)\pi -{\cos }^{-1}\frac{1}{3}$$

Commented by hardmath last updated on 19/May/24

$$\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{mych}\mathrm{professor}$$

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