Question Number 30054 by rahul 19 last updated on 15/Feb/18
Answered by ajfour last updated on 16/Feb/18
$${let}\:\:\mathrm{sin}\:{x}=\:{s} \\ $$$$\mathrm{5}{s}^{\mathrm{2}} +\mathrm{4}{s}^{\mathrm{2}} \left(\mathrm{1}−{s}^{\mathrm{2}} \right)−\mathrm{4}\left(\mathrm{1}−\mathrm{2}{s}^{\mathrm{2}} \right)\:>\:\mathrm{0} \\ $$$${or}\:\:\:\mathrm{4}\boldsymbol{{s}}^{\mathrm{4}} −\mathrm{17}\boldsymbol{{s}}^{\mathrm{2}} +\mathrm{4}\:<\:\mathrm{0} \\ $$$$\:\:\left(\mathrm{4}\boldsymbol{{s}}^{\mathrm{2}} −\mathrm{1}\right)\left(\boldsymbol{{s}}^{\mathrm{2}} −\mathrm{4}\right)\:<\:\mathrm{0} \\ $$$$\Rightarrow\:\:\frac{\mathrm{1}}{\mathrm{4}}<\:{s}^{\mathrm{2}} \:<\:\mathrm{4}\:\:\:{and}\:{we}\:{know}\:{s}^{\mathrm{2}} \:\leqslant\mathrm{1} \\ $$$${so}\:\:\:\frac{\mathrm{1}}{\mathrm{4}}<\:{s}^{\mathrm{2}} \:\leqslant\:\mathrm{1} \\ $$$$\Rightarrow\:\boldsymbol{{n}\pi}+\frac{\boldsymbol{\pi}}{\mathrm{6}}\:<\:\boldsymbol{{x}}\:<\:\left(\boldsymbol{{n}}+\mathrm{1}\right)\boldsymbol{\pi}−\frac{\boldsymbol{\pi}}{\mathrm{6}}\:\:. \\ $$
Commented by rahul 19 last updated on 16/Feb/18
$$\mathrm{thank}\:\mathrm{u}\:\mathrm{sir}. \\ $$