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Question Number 128464 by n0y0n last updated on 07/Jan/21

$${\sin }^{-1}\left(\sin 92°\right)=?$$

Commented by mr W last updated on 07/Jan/21

$$88°$$

Answered by Olaf last updated on 08/Jan/21

$$$$$$92°=\frac{\pi }{180}\times 92\mathrm{rad}=\frac{23\pi }{45}\mathrm{rad}$$$$\mathrm{\forall }x\in \mathbb{R},\arcsin \left(\sin x\right)={(-1)}^{\lfloor \frac{x}{\pi }+\frac{1}{2}\rfloor }(x-\lfloor \frac{x}{\pi }+\frac{1}{2}\rfloor \pi )$$$$\left(\mathrm{see}\mathrm{another}\mathrm{question}\mathrm{above}\right)$$$$\lfloor \frac{\frac{23\pi }{45}}{\pi }+\frac{1}{2}\rfloor \approx \lfloor \frac{91}{90}\rfloor =1$$$$\Rightarrow \arcsin \left(\sin 92°\right)={(-1)}^1(\frac{23\pi }{45}-\pi )=\frac{22\pi }{45}$$

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