Trigonometric Inequalities

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.