Question

Найдите решение тригоном.неравенства:
1) 2sin^2(x+3П/2)≥1/2
3П/2-дробь
2)ctg3x-√3≥0

Answer 1

$2sin^2(x+ cfrac{3 pi }{2}) geq cfrac{1}{2} \ sin^2(x+ cfrac{3 pi }{2}) geq cfrac{1}{4} \ left[ begin{matrix} sin(x+ cfrac{3 pi }{2}) geq cfrac{1}{2} \ sin(x+ cfrac{3 pi }{2}) leq -cfrac{1}{2} end{array}right. \ left[ begin{matrix} cfrac{ pi }{6}+2 pi k leq x+ cfrac{3 pi }{2} leq cfrac{ 5pi }{6}+2 pi k \ - cfrac{5 pi }{6}+2 pi n leq x+ cfrac{3 pi }{2} leq -cfrac{ pi }{6}+2 pi n end{array}right.$
$left[ begin{matrix} cfrac{ pi }{6}- cfrac{3 pi }{2}+2 pi k leq x leq cfrac{ 5pi }{6}- cfrac{3 pi }{2}+2 pi k \ - cfrac{5 pi }{6}- cfrac{3 pi }{2}+2 pi n leq x leq -cfrac{ pi }{6}- cfrac{3 pi }{2}+2 pi n end{array}right. \ left[ begin{matrix} - cfrac{4 pi }{3}+2 pi k leq x leq -cfrac{ 2pi }{3}+2 pi k, kin Z \ - cfrac{7 pi }{3}+2 pi n leq x leq- cfrac{5 pi }{3}+2 pi n, nin Z end{array}right.$

$mathrm{ctg}3x- sqrt{3} geq 0 \ mathrm{ctg}3x geq sqrt{3} \ pi n textless 3x leq cfrac{ pi }{6}+ pi n \ cfrac{pi n}{3} textless x leq cfrac{ pi }{18}+ cfrac{pi n}{3}, nin Z$