Question Number 115348 by bemath last updated on 25/Sep/20 | ||
$${If}\:{x}\:\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right)\:{and}\:\mathrm{2cos}\:{x}\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)+\mathrm{tan}\:^{\mathrm{2}} {x}\:<\:\mathrm{sec}\:^{\mathrm{2}} {x}\: \\ $$ $${has}\:{solution}\:{set}\:{is}\:{a}<{x}<{b}.\:{find}\:{the} \\ $$ $${value}\:{of}\:{a}+{b} \\ $$ | ||
Answered by bobhans last updated on 25/Sep/20 | ||
$$\Rightarrow\mathrm{sin}\:\mathrm{2}{x}+\mathrm{2cos}\:^{\mathrm{2}} {x}+\mathrm{tan}\:^{\mathrm{2}} {x}\:<\:\mathrm{1}+\mathrm{tan}\:^{\mathrm{2}} {x} \\ $$ $$\Rightarrow\mathrm{sin}\:\mathrm{2}{x}+\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{2}{x}\right)\:<\:\mathrm{1} \\ $$ $$\Rightarrow\mathrm{sin}\:\mathrm{2}{x}+\mathrm{cos}\:\mathrm{2}{x}\:<\:\mathrm{0} \\ $$ $${we}\:{get}\:\frac{\mathrm{3}\pi}{\mathrm{8}}\:<\:{x}\:<\:\frac{\mathrm{4}\pi}{\mathrm{8}}\:\rightarrow{then}\:{a}+{b}\:=\:\frac{\mathrm{7}\pi}{\mathrm{8}} \\ $$ | ||