Trigonometric Inequalities

A trig inequality like sin(x) > 1/2 doesn't have a single answer or even a single interval. Because the sine wave rises and falls forever, it crosses any horizontal line over and over, so the solution is a repeating pattern of intervals, one chunk per cycle. That repetition is the whole point: once you understand one cycle, every other cycle just repeats it shifted by the period.

To solve it, first find where the graph equals the threshold exactly; those crossing points are the boundaries of your solution intervals. Between consecutive boundaries, the graph is either entirely above the line or entirely below it, so you only need to test one point in each gap to know which side wins. Sketching the horizontal line through the threshold value on top of the wave makes this visual: wherever the curve pokes above the line, shade that stretch of the x-axis.

This topic pulls together graphing transformations and inverse trig values from earlier in the unit. You need the shape and period of the graph to know how many intervals to expect, and you need arcsin, arccos, or arctan to pin down the exact boundary numbers. It also sets up later work where trig inequalities model real constraints, like when a Ferris wheel's height or a sound wave's amplitude must stay above or below some limit.