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Find the solution of $⌊{\mathrm{sin}}^{-1}\left(x\right)⌋>⌊{\mathrm{cos}}^{-1}\left(x\right)⌋$
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pheniankang
Noting that $⌊\mathrm{arcsin}x⌋=-2,-1,0,1$ and that $⌊\mathrm{arccos}x⌋=0,1,2,3$ we have
$\begin{array}{rl}⌊\mathrm{arcsin}x⌋>⌊\mathrm{arccos}x⌋& \phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}⌊\mathrm{arcsin}x⌋=1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}⌊\mathrm{arccos}x⌋=0\\ & \phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}1\le \mathrm{arcsin}x<2\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}0\le \mathrm{arccos}x<1\\ & \phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\mathrm{sin}1\le x\le 1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{cos}1