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Question Number 122337 by benjo_mathlover last updated on 15/Nov/20

$$Finddomainoffunction$$$$r\left(x\right)=\mathrm{arc}\cos \left(\frac{3}{4+2\sin x}\right)$$

Commented by liberty last updated on 16/Nov/20

$$\mathrm{The}\mathrm{function}\mathrm{defined}\mathrm{for}\mathrm{values}\mathrm{of}\mathrm{x}$$$$\mathrm{for}\mathrm{which}-1⩽\frac{3}{4+2\sin \mathrm{x}}⩽1.$$$$\mathrm{Since}4+2\sin \mathrm{x}>0\mathrm{at}\mathrm{any}\mathrm{x},\mathrm{the}\mathrm{problem}$$$$\mathrm{is}\mathrm{reduced}\mathrm{to}\mathrm{solving}\mathrm{inequality}$$$$\frac{3}{4+2\sin \mathrm{x}}⩽1,\mathrm{whence}3⩽4+2\sin \mathrm{x}$$$$\mathrm{i}.\mathrm{e}\sin \mathrm{x}⩾-\frac{1}{2}\Rightarrow -\frac{\pi }{6}+2\mathrm{k}\pi ⩽\mathrm{x}⩽\frac{7\pi }{6}+2\mathrm{k}\pi$$$$\mathrm{for}\mathrm{k}ϵ\mathbb{Z}.▴$$

Commented by benjo_mathlover last updated on 16/Nov/20

Commented by benjo_mathlover last updated on 16/Nov/20

$$correct..thanks$$

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