Elementary inequality with the tangent

Problem

Solve the inequality: $\tan x > 1$

Solution

We have $\tan x = 1$ at $x=\dfrac{\pi}{4}$. On the axis of tangents, $\tan x$ exceeds $1$ on the arcs where the point $P_x$ lies between $\dfrac{\pi}{4}$ and the asymptote $\dfrac{\pi}{2}$ (where the tangent is not defined). On $[0,2\pi)$, taking the opposite branch into account: $x\in\left(\frac{\pi}{4},\frac{\pi}{2}\right)\cup\left(\frac{5\pi}{4},\frac{3\pi}{2}\right)$ Since the tangent has period $\pi$, the solution is written in compact form: $\boxed{\;x\in\left(\frac{\pi}{4}+k\pi,\ \frac{\pi}{2}+k\pi\right)\;}$

Links

Topics: Trigonometric inequalities
Concepts: Trigonometric inequality · Trigonometric inequality reading the circle
Functions: Tangent
Skills: Interpreting a graph · Solving inequalities
Exercise type: Solving an inequality