Question

Помогите решить примеры:
1) sin7x=sinx
2) √2sinx/2≥1
3) (sinx+cosx)^2=cos2x
4) 3sin2x+4cos2x=5 

Answer 1

Решение  1,3,4  во вложении.

Answer 2

$2.quad sqrt2sinfrac{x}{2} geq 1\\sinfrac{x}{2} geq frac{1}{sqrt2}\\frac{pi}{4}+2pi n leq frac{x}{2} leq frac{3pi }{4}+2pi n\\frac{pi}{2}+4pi n leq x leq frac{3pi }{2}+4pi n,quad nin Z$

$4.quad 3sin2x+4cos2x=5; |:5; ; (5=sqrt{3^2+4^2}=sqrt{25})\\frac{3}{5}sin2x+frac{4}{5}cos2x=1\\(; ; (frac{3}{5})^2+(frac{4}{5})^2=1; ; to frac{3}{5}=sin alpha ,; frac{4}{5}=cos alpha to tg alpha =frac{3}{4},alpha =arctgfrac{3}{4})\\sin alpha cdot sin2x+cos alpha cdot cos2x=1\\sin( 2x+ alpha )=1\\2x+ alpha =frac{pi}{2}+2pi n\\2x=frac{pi}{2}- alpha +2pi n\\x=frac{pi}{4}-frac{ alpha }{2}+pi n,; nin Z\\x=frac{pi}{4}-frac{arctgfrac{3}{4}}{2}+pi n$