Question

Помогите срочно с тригонометрическими уравнениями!!! Даю 100 баллов.

Answer 1

$1) 5cos^2frac{2x}{3}-10 cosfrac{2x}{3} =0|:5\\cos^2frac{2x}{3}-2cosfrac{2x}{3}=0;\\cosfrac{2x}{3}(cosfrac{2x}{3}-2)=0Rightarrow cosfrac{2x}{3}=0\\frac{2x}{3}=frac{pi}{2}+ pi n,nin mathbb Z|cdotfrac{3}{2} \\x=frac{3pi}{4}+frac{3pi n}{2},nin mathbb Z.$

$\\2)sin(4pi+frac{5x}{4})=sin(frac{5pi}{2}-frac{5x}{4});\\ sinfrac{5x}{4}=cos frac{5x}{4} |:cos frac{5x}{4}neq 0\\frac{sinfrac{5x}{4}}{cos frac{5x}{4}} =1,\\tgfrac{5x}{4}} =1\\frac{5x}{4}}=arctg(1)+pi n,ninmathbb Z;\\frac{5x}{4}}=frac{pi}{4}+ pi n,ninmathbb Z|cdotfrac{4}{5} \\x=frac{pi}{5} +frac{4pi n}{5}, ninmathbb Z$

$3)tg^2(2pi-3x)+2ctg(frac{pi}{2}+3x)-3=0;\\ (-tg3x)^2-2tg3x-3=0;\\tg^23x-2tg3x-3=0Rightarrow\\Rightarrowleft [ {{tg3x=-1,} atop {tg3x=3;}} right. left [ {{3x=arctg(-1)+pi n, ninmathbb Z} atop {3x=arctg(3)+pi n, ninmathbb Z|:3}} right. \\left [{{3x=-frac{pi}{4}+pi n, ninmathbb Z |:3} atop {x=frac{1}{3}arctg3+frac{pi n}{3},ninmathbb Z }} right.$

$left [ {{x=-frac{pi}{12}+frac{pi n}{3},ninmathbb Z } atop {x=frac{1}{3}arctg3+frac{pi n}{3},ninmathbb Z} right.$