Trigonometric inequalities and graph transformations

Inequalities of the form a₁ < sin x < b₁, a₂ < cos x < b₂, a₃ < tan x < b₃ are the simplest trigonometric inequalities. The value sin x is the ordinate of a point on the unit circle and cos x is its abscissa. To solve sin x > a we find the arc of points with ordinate greater than a, taking its ends from the equation sin x = a. For cos x > a we take the arc whose abscissa exceeds a. We solve first within one period (say [0, 2π]) and then add the period: 2πk for sin and cos, πk for tan, k ∈ ℤ. If the inequality is strict the endpoints are excluded; if not, they are included. To check, pick one point of the solution and substitute it into the inequality.