Question

Решите тригонометрические неравенства.(c помощью единичной окружности)
а)$cos frac{2x}{3} textless - frac{ sqrt{2} }{2}$
б)$2sin frac{x}{4} cos frac{x}{4} textgreater - frac{1}{2}$

Answer 1

1)3π/4+2πn<2x/3<5π/4+2πn
9π/8+3πn<x<15π/8+3πn
2)2sinx/4*cosx/4=sinx/2>-1/2
-π/6+2πn<x/2<7π/6+2πn
-π/3+4πn<x<7π/3+4πn

Answer 2

а) cos2x/3 < -√2/2 ;
π -π/4 < 2x/3 < π +π/4 ;
3π/4 <2x/3 <5π/4 ;
2πk+3π/4 <2x/3 <5π/4 +2πk ;
3πk +9π/8 < x <15π/8 +3πk , k∈ Z.
б) 2sinx/4cosx/4 > -1/2 '
sin2*x/4 > -1/2;
 - π/6 < x/2 < π +π/6;
 - π/6 < x/2 < 7π/6;
2πk  - π/6 < x/2 <7π/6  +2πk ;
4πk -π/3 < x <  7π/3 +4πk , k∈ Z
- π/3 +4πk  < x <  7π/3 +4πk , k∈ Z.
в любом  интервале
x ∈(- π/3 +4πk ;  7π/3 +4πk )  , k∈ Z.