Question

99 БАЛЛОВ!!! ПОМОГИТЕ!!!! ТРИГОНОМЕТРИЯ!
10 sin^2(x/3)-12 sin(x/3)cos(/3)-11 cos^2(x/3)=1
cos(3π/2+x)-5cosx=0
√3ctg(π/3-4x)=3

Answer 1

$1); ; 10sin^2 frac{x}{3} -12sin frac{x}{3} cdot cosfrac{x}{3} -11cos^2frac{x}{3} =1\\ 10sin^2frac{x}{3}-12sin frac{x}{3} cdot cosfrac{x}{3} -11cos^2frac{x}{3}=sin^2frac{x}{3} +cos^2frac{x}{3} \\9sin^2 frac{x}{3} -12sin frac{x}{3}cdot cos frac{x}{3} -12cos^2 frac{x}{3} =0; |:3cos^2 frac{x}{3} ne 0\\3tg^2 frac{x}{3} -4tgfrac{x}{3} -4=0\\D/4=4+12=16$

$a); ; tgfrac{x}{3} = frac{2-4}{3} =- frac{2}{3} \\ frac{x}{3}=-arctgfrac{2}{3}+pi n; ,; nin Z\\underline {x=-3arctgfrac{2}{3}+3pi n,; nin Z}\\b); ; tg frac{x}{3}= frac{2+4}{3}=2$

$frac{x}{3}=arctg2+pi k,; kin Z\\underline {x=3arctg2+3pi k,; kin Z}$

$2); ; cos( frac{3pi }{2} +x)-5cosx=0\\sinx-5cosx=0; |:cosxne 0\\tgx-5=0\\tgx=5\\underline {x=arctg5+pi n,; nin Z}\\3); ; sqrt3cdot ctg(frac{pi}{3}-4x)=3\\ctg( frac{pi }{3} -4x)=sqrt3\\ frac{pi }{3} -4x=arcctgsqrt3+pi n; ,; nin Z\\ frac{pi }{3} -4x= frac{pi }{6}+pi n,; nin Z\\4x= frac{pi }{3}- frac{pi }{6} -pi n=frac{pi}{6}-pi n,; nin Z\\underline {x=frac{pi}{24}-frac{pi n}{4}; ,; nin Z}$

$(Mozno; pisat; tak:; ; x=frac{pi}{24}+frac{pi n}{4}; ,; nin Z)$