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Question Number 128546 by mr W last updated on 08/Jan/21

$$findasimpleexpressionfor$$$$y={\sin }^{-1}\left(\sin x\right)$$$$x\in \mathbb{R}andxinrad.$$

Commented by liberty last updated on 08/Jan/21

$$\sin \mathrm{x}=\sin \mathrm{y};\mathrm{y}=\mathrm{x}+2\mathrm{k}\pi$$$$\mathrm{where}-\frac{\pi }{2}<\mathrm{y}<\frac{\pi }{2}$$$$$$

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

$$example:$$$$x=5$$$$howdoyougety=f\left(5\right)withyour$$$$formula?$$

Commented by john_santu last updated on 08/Jan/21

$$\mathrm{do}\mathrm{you}\mathrm{meant}5\mathrm{in}\mathrm{degress}?\mathrm{or}\mathrm{radiant}-?$$

Commented by Olaf last updated on 08/Jan/21

$$f\left(5\right)={(-1)}^{\lfloor \frac{5}{\pi }+\frac{1}{2}\rfloor }(5-\lfloor \frac{5}{\pi }+\frac{1}{2}\rfloor \pi )$$$$\lfloor \frac{5}{\pi }+\frac{1}{2}\rfloor \approx \lfloor 2,091\rfloor =2$$$$\Rightarrow f\left(5\right)={(-1)}^2(5-2\pi )=5-2\pi$$

Answered by Olaf last updated on 08/Jan/21

$$\mathrm{\forall }x\in \mathbb{R},\mathrm{\exists }k\in \mathbb{Z}\mathrm{\setminus }x-k\pi \in [-\frac{\pi }{2},+\frac{\pi }{2}[$$$$\iff k\pi ⩽x+\frac{\pi }{2}<(k+1)\pi$$$$\iff k⩽\frac{x}{\pi }+\frac{1}{2}<k+1$$$$k=\lfloor \frac{x}{\pi }+\frac{1}{2}\rfloor \left(\mathrm{integer}\mathrm{part}\right)$$$$\mathrm{and}\mathrm{\forall }\theta \in \mathbb{R},\mathrm{\forall }q\in \mathbb{Z},\sin (\theta +q\pi )={(-1)}^q\sin \theta$$$$\Rightarrow \sin x=\sin (x-k\pi +k\pi )={(-1)}^k\sin (x-k\pi )$$$$\Rightarrow \arcsin \left(\sin x\right)=\arcsin \left({(-1)}^k\sin (x-k\pi )\right)$$$$={(-1)}^k\arcsin \left(\sin (x-k\pi )\right)$$$$\mathrm{and}x-k\pi \in [-\frac{\pi }{2},+\frac{\pi }{2}[$$$$\Rightarrow \arcsin \left(\sin x\right)={(-1)}^k(x-k\pi )$$$$$$$$\mathbf{Formula}\mathbf{for}\mathbf{arcsin}\left(\mathbf{sin}\mathit{x}\right):$$$$\arcsin \left(\sin x\right)={(-1)}^{\lfloor \frac{x}{\pi }+\frac{1}{2}\rfloor }(x-\lfloor \frac{x}{\pi }+\frac{1}{2}\rfloor \pi )$$

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

$$thanksalot!$$

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