Question

Решите неравенство
$sin^2 x-6sin xcos x+5cos^2 x textgreater 0;\ 1+2sin x geq 4sin x cos x+2cos x$

Answer 1

$1)quad sin^2x-6sinxcdot cosx+5cos^2x textgreater 0; |:cos^2x textgreater 0,cosxne 0\\tg^2x-6tgx+5 textgreater 0\\(tgx)_1=1; ; ili; ; (tgx)_2=5; ,qquad +++(1)---(5)+++\\tgxin (-infty ,1)cup (5,+infty )\\ left [ {{tgx textless 1} atop {tgx textgreater 5}} right. ; left [ {{x_1in (-frac{pi}{2}+pi n,; frac{pi}{4}+pi n)} atop {x_2in (arctg5+pi n,; frac{pi}{2}+pi n)}} right. \\xin (-frac{pi}{2}+pi n,; frac{pi}{4}+pi n)cup (arctg5+pi n,frac{pi}{2}+pi n); ,nin Z$

$2)quad 1+2sinx geq 4sinxcdot cosx+2cosx\\2(sinx-cosx)-4sinxcdot cosx+1 geq 0\\t=sinx-cosx; ; to ; ; t^2=(sinx-cosx)^2=1-2sinxcdot cosx; to \\2sinxcdot cosx=1-t^2; ; to \\2t-2(1-t^2)+1 geq 0\\2t^2+2t-1 geq 0\\D/4=1+2=3; ,; ; t_{1,2}= frac{-1pm sqrt3}{2}\\+++( frac{-1-sqrt3}{2} )---( frac{-1+sqrt3}{2} )+++ \\tin (-infty , frac{-1-sqrt3}{2} , ]cup[, frac{-1+sqrt3}{2} ,+infty )\\a); ; sinx-cosx leq frac{-1-sqrt3}{2} |:sqrt2$

$frac{1}{sqrt2}sinx- frac{1}{sqrt2} cosx leq frac{-1-sqrt3}{2sqrt2} \\cosfrac{pi}{4}cdot sinx-sinfrac{pi}{4}cdot cosx leq frac{-1-sqrt3}{2sqrt2} \\sin(x-frac{pi}{4} )leq frac{-1-sqrt3}{2sqrt2} ; ,; ; frac{-1-sqrt3}{2sqrt2} approx -0,97$

$arcsin frac{-1-sqrt3}{2sqrt2} +2pi nleq x-frac{pi}{4} leq pi +arcsin frac{-1-sqrt3}{2sqrt2} +2pi n; ,; nin Z$

$frac{pi}{4}+arcsin frac{-1-sqrt3}{2sqrt2} +2pi n leq x leq frac{5pi}{4} +arcsin frac{-1-sqrt3}{2sqrt2}+2pi n$

$b)quad sinx-cosx geq frac{-1+sqrt3}{2} ; |:sqrt2\\sin(x-frac{pi}{4}) geq frac{-1+sqrt3}{2sqrt2}; ,; ; frac{-1+sqrt3}{2sqrt2} approx 0,26\\ pi -arcsin frac{-1+sqrt3}{2sqrt2} +2pi nleq x-frac{pi}{4} leq arcsin frac{-1+sqrt3}{2sqrt2} +2pi n,; nin Z\\ frac{5pi}{4}-arcsin frac{-1+sqrt3}{2sqrt2} +2pi nleq x leq frac{pi}{4}+arcsinfrac{-1+sqrt3}{2sqrt2}+2pi n$

$Otvet:; xin [, frac{pi}{4}+arcsin frac{-1-sqrt3}{2sqrt2} +2pi n,frac{5pi}{4}+arcsin frac{-1-sqrt3}{2sqrt2} +2pi n]cup$

$cup [, frac{5pi}{4} -arcsin frac{-1+sqrt3}{2sqrt2}+2pi n,; frac{pi}{4} +arcsin frac{-1+sqrt3}{2sqrt2} +2pi n, ]$