Equations - Trigonometric Inequation Sin

Trigonometric Inequation – Sine

Definition and Case-wise Solutions

A trigonometric inequation involving sine compares \( \sin x \) to a constant \( m \). The inequality has different solution scenarios depending on the value of \( m \) in relation to the sine function's range \([-1, 1]\).

Graphical explanation of sine inequation

Key Inequation

\[ \sin x \geq m \]

The nature of the solution 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:
    \[ -\alpha + 2k\pi \leq x \leq \alpha + 2k\pi \] where \( \alpha = \arccos m \), and \( 0 \leq \alpha \leq \pi \).

Applications

  • Used in solving inequalities involving wave functions.
  • Important in signal processing and harmonic motion analysis.
  • Helpful in modeling oscillations and periodic constraints in physics and engineering.
×

×