Elementary inequalities with the tangent

For an inequality with the tangent, of the type $\tan x \ge k$, one uses the graph of the tangent or the line “tangent” to the circle at the point $(1;0)$: the axis of tangents, that is the vertical line $x=1$.

On this axis the value $\tan x$ is read as the ordinate of the point where the line $OP_x$ (extended) meets the line $x=1$.

Reading for $\tan x > 1$: we have $\tan x = 1$ at $\dfrac{\pi}{4}$ (arctangent of $1$); the inequality is satisfied on the green arcs, where the point of intersection with the axis of tangents has ordinate greater than $1$.

Since the tangent has period $\pi$ (not $2\pi$), the solution is generalised by adding $k\pi$: the condition repeats twice for every complete turn of the circle.

Links

Topics: Trigonometric inequalities
Concepts: Unit circle · Trigonometric inequality · Trigonometric inequality reading the circle
Functions: Tangent
Methods: Trigonometric inequalities
Skills: Interpreting a graph · Solving inequalities