Solving 2sin(x) - 1 > 0

Trigonometry · Axiom Academy

EXAMPLE Solving Trigonometric Inequalities Master the process of finding solution intervals using critical values, test points, and interval notation. Excellent work! You've mastered solving trigonometric inequalities. Here's what we learned: Isolate the Trig Function: First, use algebra to isolate the trigonometric function (sin(x), cos(x), etc.) on one side of the inequality. Find Critical Values: Solve the related equation (with = instead of > or <) to find where the function equals the boundary value. These critical points divide the domain into testable intervals. Create Test Intervals: The critical values split your domain into separate regions. For [0, 2π), two critical values create three intervals to test. Test Representative Points: Choose a convenient value from each interval and check if it satisfies the original inequality. The entire interval has the same truth value as your test point. Interval Notation Matters: Use parentheses ( ) for strict inequalities ( ) and brackets [ ] for inclusive inequalities (≤ or ≥). The endpoints (critical values) are included only when the inequality includes equality. General Solutions: For periodic functions, add the period (2π for sine and cosine) multiplied by any integer n to express all solutions. This systematic approach works for all trigonometric inequalities. Remember: critical values → test intervals → check endpoints → write solution!

This is the written version of the interactive lesson above. See the full Trigonometry course.