Trigonometric Inequalities — Complete Guide
What is a trigonometric inequality? A trigonometric inequality is an inequality in which the unknown appears as the argument of one or more trigonometric functions. Unlike algebraic inequalities, the solution set is typically an infinite union of intervals, because trigonometric functions are periodic.
The answer to a trigonometric inequality is never a set of isolated angles — it is a union of intervals, continuous stretches of the real line where the condition holds. Because the trig functions repeat, these intervals repeat with the period of the function.
1. Two Primary Solution Methods
Unit Circle Method
- Solve the corresponding equation to find boundary angles.
- On the unit circle, identify the arc where the coordinate condition holds (e.g., where the $y$-coordinate exceeds $k$ for $\sin x > k$).
- Write the arc as an interval, traversing anticlockwise.
- Add the period ($+2n\pi$ for sin/cos, $+n\pi$ for tan/cot) to produce the full family.
Graphing Method
- Sketch the graph of the trig function over one or two periods.
- Draw the horizontal line $y = k$.
- Read off the $x$-intervals where the curve lies above (or below) the line.
- Extend by periodicity.
Both methods yield the same result; the choice depends on whether a visual or geometric perspective is more natural for the problem at hand.
2. Elementary Sine Inequalities
Let $\alpha = \arcsin k$ where $|k| \le 1$, so $\alpha \in [-\tfrac{\pi}{2},\,\tfrac{\pi}{2}]$.