Ответ:

1) sin2x >/ 0

arcsin(0) + 2πk </ 2x </ π-arcsin(0) + 2πk , k € Z

0 + 2πk </ 2x </ π-0 + 2πk , k € Z

2πk </ 2x </ π + 2πk , k € Z / : 2

πk </ x </ π/2 + πk , k € Z

2) cos x/2 < √2/2

arccos(√2/2) + 2πk < x/2 < 2π - arccos(√2/2) + 2πk , k € Z

π/4 + 2πk < x/2 < 2π - π/4 + 2πk , k € Z

π/4 + 2πk < x/2 < 8π/4 - π/4 + 2πk , k € Z

π/4 + 2πk < x/2 < 7π/4 + 2πk , k € Z / : 1/2

π/2 + 4πk < x < 7π/2 + 4πk , k € Z

3) tg ( x - π/3 ) > √3

Замена (x-π/3) = a

tg a > √3

arctg(√3) + πk > a > π/2 + πk , k € Z

π/3 + πk > x - π/3 > π/2 + πk , k € Z

π/3 + π/3 + πk > x > π/2 + π/3 + πk , k € Z

2π/3 + πk > x > 3π/6 + 2π/6 + πk , k € Z

2π/3 + πk > x > 5π/6 + πk , k € Z

4) ctg (x+π/3) </ - 1/√3

Замена (x+π/3) = a

ctg a </ - 1/√3

arcctg(-1/√3) + πk </ a </ π + πk , k € Z

(π-arcctg(1/√3)) + πk </ a </ π + πk , k € Z

(π-π/3) + πk </ a </ π + πk , k € Z

2π/3 + πk </ a </ π + πk , k € Z

2π/3 + πk </ x+π/3 </ π + πk , k € Z

2π/3 - π/3 + πk </ x </ π - π/3 + πk , k € Z

π/3 + πk </ x </ 2π/3 + πk , k € Z