Maths: Solving Cosine Inequalities for River Depth

Trigonometric modelling turns the steady rise and fall of real-world cycles — tides, temperatures, daylight hours — into equations you can analyse precisely. At its heart lies the cosine wave, whose values oscillate between -1 and 1, so a function like d(t) = 5 - 4cos(πt/6) shifts and stretches that wave to describe water depth over a 24-hour period. The constant term sets the midline, while the amplitude determines how far the depth swings above and below it. Understanding this structure matters because it lets you predict extremes and durations without ever measuring the river directly. The maximum occurs where cos(πt/6) = -1, and the minimum where it equals 1, with the period 2π/(π/6) = 12 hours governing how often the cycle repeats. Solving inequalities such as cos(πt/6) > 1/2 then reveals the intervals when conditions stay within a required range, connecting the algebra of the wave to practical, time-bound decisions.