Question Number 155878 by mnjuly1970 last updated on 05/Oct/21
$$ \\ $$$$\:\:{the}\:{equation}\:{of}\:: \\ $$$$\:\:\:\:\left({a}\:+{sin}\left({x}\right)\right)\left({a}+{cos}\left({x}\right)\right)={a} \\ $$$$\:\:\:\:{has}\:\:{solution}\:{in}\:\mathbb{R}\: \\ $$$$\:\:\:\:\:\:\:{hence}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{a}\:\in\:? \\ $$$$\:\:\:\: \\ $$$$ \\ $$$$ \\ $$
Answered by MJS_new last updated on 05/Oct/21
$$\mathrm{solve}\:\mathrm{it}\:\mathrm{for}\:{a}\:\mathrm{and}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}: \\ $$$$\mathrm{1}−\sqrt{\mathrm{2}}\leqslant{a}\leqslant\frac{\mathrm{1}+\sqrt{\mathrm{2}}+\sqrt{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$
Commented by talminator2856791 last updated on 05/Oct/21
$$\:\mathrm{did}\:\mathrm{you}\:\mathrm{also}\:\mathrm{use}\:\mathrm{wolframalpha}? \\ $$
Commented by MJS_new last updated on 05/Oct/21
$$\mathrm{no}.\:\mathrm{I}\:\mathrm{used}\:\mathrm{my}\:\mathrm{own}\:\mathrm{brain}. \\ $$
Answered by talminator2856791 last updated on 05/Oct/21
$$\:{a}\:\in\:\left(\mathrm{1}−\sqrt{\mathrm{2}},\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}}\right)\:\: \\ $$