Trigonometric Inequality Solver

Solving Trigonometric Inequalities

Where a trigonometric equation like sin(x) = 0.5 asks for specific angles, an inequality like sin(x) ≥ 0.5 asks for an entire range: every x for which the curve sits at or above the threshold k. Because sine, cosine, and tangent repeat forever, that range is never a single interval — it is the same pattern of intervals copied every period, stretching across the whole number line.

Within one period, sine and cosine each rise to a peak and fall back to a trough exactly once, so wherever they cross the value k, they do it at exactly two points, and the inequality holds on the single contiguous arc between them (or outside it, for the opposite direction). Tangent behaves differently: it climbs from −∞ to +∞ between two consecutive vertical asymptotes, crossing every value of k exactly once, so its satisfying region is a single interval bounded on one side by that crossing point and on the other by an asymptote, where the function is undefined.

Finding that base interval, in the variable u = ax + b, uses the same inverse functions as solving an equation — arcsin, arccos, or arctan locate the crossing point(s), and the shape of the sine, cosine, or tangent curve determines which side of those points satisfies the inequality. The last step undoes the substitution, x = (u − b) / a: if a is positive the interval keeps its order, and if a is negative the endpoints swap places, since dividing by a negative number reverses which one is smaller.

Once this base interval is known, the calculator repeats it every x-period — 2π / |a| for sine and cosine, π / |a| for tangent — to build the full, infinite solution set, and narrows it to a concrete list of intervals whenever you enter a bounding range. The graph makes the same picture visual: the green curve traces fn(ax + b), the dashed line marks the threshold k, and the shaded bands show exactly where the curve satisfies the inequality across the plotted window.