Equations - Trigonometric Inequation Cos

Trigonometric Inequation – Cosine

Definition and Case-wise Solutions

A trigonometric inequation involving cosine compares \( \cos x \) to a constant \( m \). The nature of the solution depends on the value of \( m \), particularly whether it lies within the valid range of the cosine function, which is \([-1, 1]\).

Graphical explanation of cosine inequation

Key Inequation

\[ \cos x \geq m \]

The solution set depends on the value of \( m \):

  • If \( m < -1 \): The inequality is always true for all \( x \).
  • If \( m > 1 \): There is no solution.
  • If \( |m| \leq 1 \): The solution is given by:
    \[ \alpha + 2k\pi \leq x \leq (\pi - \alpha) + 2k\pi \] where \( \alpha = \arcsin m \), and \( -\frac{\pi}{2} \leq \alpha \leq \frac{\pi}{2} \).

Applications

  • Solving periodic inequalities in trigonometry.
  • Used in physics problems involving wave motion and phase shifts.
  • Helpful in modeling constraints in engineering and signal processing.
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