Periodic threshold problems and inequalities

Periodic thresholds and trigonometric inequalities

Find the times when an idealized periodic quantity is above, below or at a specified level. This precalculus practice set connects sinusoidal equations with interval reasoning, using tide-style levels, repeating signals and a labelled threshold graph. Each problem specifies a meaningful time window or angle interval.

Questions progress from reading crossings and interpreting inequalities to repeated cycles, phase shifts, negative coefficients and nonspecial inverse values. Single-select and multiselect tasks distinguish isolated touches at an extreme from intervals of time above a level. Explanations show how to find boundary points, test which side satisfies the inequality and apply the original endpoints.

Checking a threshold solution

Review scaled-angle equations, sinusoidal models and interval notation. First compare the threshold with the function’s range: the condition may be impossible or true throughout the domain. Then identify crossings or tangencies and decide whether equality is included. When calculating duration, add the lengths of disjoint intervals without counting overlapping cycles twice. The models here are supplied mathematical assumptions, not operational forecasts for real tides or machinery.