1) cos 2x < 0

cos 2x < 0
cosine function is negative in the second and third quadrants
so, we have:
π < 2x < 2π/3 or 4π/3 < 2x < 2π

Dividing by 2:
π/2 < x < π/3 or 2π/3 < x < π

So the solution is:
π/2 < x < π/3 or 2π/3 < x < π

2) sin x > 1/4

sin x > 1/4
sine function is positive in the first and second quadrants
so, we have:
0 < x < π/2 or π < x < 3π/2

So the solution is:
0 < x < π/2 or π < x < 3π/2