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Question Number 140497 by benjo_mathlover last updated on 08/May/21

$${\sin }^{-1}\left(\sin \mathrm{x}\right)=\mathrm{x}$$$${\sin }^{-1}\left(\cos \mathrm{x}\right)=?$$$${\sin }^{-1}\left(\tan \mathrm{x}\right)=?$$

Commented by mr W last updated on 08/May/21

$${\sin }^{-1}\left(\sin \mathrm{x}\right)=\mathrm{x}isnotalwaystrue.$$

Commented by liberty last updated on 09/May/21

$$\mathrm{for}\mathrm{what}\mathrm{value}\mathrm{of}\mathrm{x}\mathrm{this}\mathrm{not}\mathrm{true}?$$

Commented by mr W last updated on 09/May/21

$$example$$$${\sin }^{-1}\left(\sin \frac{5\pi }{4}\right)=-\frac{\pi }{4}\ne \frac{5\pi }{4}$$

Commented by liberty last updated on 09/May/21

$$\frac{5\pi }{4}\notin \mathrm{D}\mathrm{since}\mathrm{not}\mathrm{satisfy}\mathrm{the}\mathrm{equation}$$

Commented by liberty last updated on 09/May/21

$$\mathrm{y}={\sin }^{-1}\left(\mathrm{x}\right)\mathrm{has}\mathrm{domain}-\frac{\pi }{2}⩽\mathrm{x}⩽\frac{\pi }{2}$$

Commented by mr W last updated on 09/May/21

$$\mathrm{y}={\sin }^{-1}\left(\mathrm{x}\right)\mathrm{has}\mathrm{domain}-1⩽\mathrm{x}⩽1$$$$\mathrm{y}={\sin }^{-1}\left(\mathrm{x}\right)\mathrm{has}range-\frac{\pi }{2}⩽\mathrm{x}⩽\frac{\pi }{2}$$$$$$$$\mathrm{y}={\sin }^{-1}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{has}\mathrm{domain}-\mathrm{\infty }⩽\mathrm{x}⩽\mathrm{\infty }$$$$\mathrm{y}={\sin }^{-1}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{has}range-\frac{\pi }{2}⩽\mathrm{x}⩽\frac{\pi }{2}$$

Commented by mr W last updated on 09/May/21

Answered by EDWIN88 last updated on 08/May/21

$$\mathrm{let}{\sin }^{-1}\left(\cos \mathrm{x}\right)=\mathrm{y}$$$$\cos \mathrm{x}=\sin \mathrm{y};\cancel{\sin }(\frac{\pi }{2}-\mathrm{x})=\cancel{\sin }\mathrm{y}$$$$\mathrm{we}\mathrm{get}\mathrm{y}=\frac{\pi }{2}-\mathrm{x}\mathrm{so}{\sin }^{-1}\left(\cos \mathrm{x}\right)=\frac{\pi }{2}-\mathrm{x}$$

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