We will consider here the simplest trigonometric inequality of the form f(x)<a (or f(x)>a), where f(x) is some trigonometric function.
Example. Solve `cos(x)<0.5`.
Let's draw graph of the function `y=cos(x)` and draw line `y=0.5`. We are interested in those intervals where cosine is below line y=0.5. One of such intervals is `(pi/3,(5pi)/3)`. Using periodicity of cosine we obtain solution: `pi/3+2pik<x<(5pi)/3+2pik,k in Z`.