Equations - Trigonometric Inequation Tan

Trigonometric Inequation – Tangent

Definition and Solution Interval

A trigonometric inequation involving tangent compares \( \tan x \) to a constant \( m \). The solution relies on the periodic nature of the tangent function and its principal values defined within its valid range.

Graphical explanation of tangent inequation

Key Inequation

\[ \tan x \geq m \]

For values of \( m \in \mathbb{R} \), the solution interval is:

\[ \alpha + k\pi \leq x \leq \frac{\pi}{2}(2k+1) \]

where \( \alpha = \arctan m \), and \( -\frac{\pi}{2} \leq \alpha \leq \frac{\pi}{2} \).

Applications

  • Used to solve angular constraints in engineering and robotics.
  • Appears in navigation, trigonometric modeling, and angular motion problems.
  • Key to solving inequalities involving tangent in advanced trigonometric equations.
×

×