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Question Number 174071 by alcohol last updated on 24/Jul/22

Answered by aleks041103 last updated on 24/Jul/22

$$.$$$$1.$$$$F\left(x\right)={\int }_0^x\frac{dt}{\sqrt{1-t^2}}={\int }_0^{x}F{}^{\prime }\left(t\right)dt$$$$\Rightarrow F{}^{\prime }\left(x\right)=\frac{1}{\sqrt{1-x^2}}$$$$2.F\left(x\right)={\int }_0^x\frac{dt}{\sqrt{1-t^2}}=arcsin\left(t\right){\mid }_0^x$$$$F\left(x\right)=arcsin\left(x\right)\Rightarrow G\left(x\right)=-arcsin\left(x\right)$$$$F\left(sin\left(x\right)\right)=arcsin\left(sin\left(x\right)\right)=x$$$$3.$$$$G\left(cos\left(x\right)\right)=-arcsin\left(cos\left(x\right)\right)$$$$-arcsin\left(t\right)=arccos\left(t\right)-\pi /2$$$$\Rightarrow G\left(cos\left(x\right)\right)=x-\frac{\pi }{2}$$$$4.Fisaninversetothesin$$$$Gisnotaninverseforthecos$$

Answered by CElcedricjunior last updated on 25/Jul/22

$$\frac{\mathit{r}\mathit{e}\mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}{}$$$$1)\mathit{F}{}^{\prime }\left(\mathit{x}\right)=\frac{1}{\sqrt{1-{\mathit{x}}^2}}-1$$$$2)\mathit{F}\left(\mathit{s}\mathit{i}\mathit{n}\left(\mathit{x}\right)\right)={\int }_0^{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}\frac{\mathit{d}\mathit{t}}{\sqrt{1-{\mathit{t}}^2}}$$$$\mathit{p}\mathit{o}\mathit{s}\mathit{o}\mathit{n}\mathit{s}\mathit{t}=\mathit{s}\mathit{i}\mathit{n}\mathit{a}$$$$\mathit{d}\mathit{t}=\mathit{c}\mathit{o}\mathit{s}\mathit{a}\mathit{d}\mathit{a}$$$$\mathit{q}\mathit{d}:\{\begin{array}{l}\mathit{t}=\mathit{s}\mathit{i}\mathit{n}\mathit{x}\\ \mathit{t}=0\end{array}=>\{\begin{array}{l}\mathit{a}=\mathit{x}\\ \mathit{t}=0\end{array}$$$$\mathit{F}\left(\mathbf{sinx}\right)={\int }_0^{\mathit{x}}\frac{\mathit{c}\mathit{o}\mathit{s}\mathit{a}\mathit{d}\mathit{a}}{\sqrt{-\mathit{s}\mathit{i}\stackrel{2}{\mathit{n}\mathit{x}}}}={\int }_0^{\mathit{x}}\mathit{d}\mathit{a}=\mathit{x}$$$$\mathbf{F}\left(\mathbf{sinx}\right)=\mathbf{x}$$$$3)\mathbf{G}\left(\mathbf{cosx}\right)=-\mathbf{x}+\mathit{\pi }/2$$$$$$

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