Maths: Reversing a Sine Model to Find Time

Sinusoidal functions are among the most powerful tools for modelling anything that repeats with a regular rhythm — tides, temperatures, sound waves, and the depth of water in a harbour. Any such model can be written in the form h(t) = A sin(Bt) + C, where the amplitude A sets how far the value swings above and below its centre, the vertical shift C fixes that centre line, and B controls how quickly the cycle repeats. Reading these parameters directly from an equation lets you state the maximum and minimum at once: the maximum is C + A and the minimum is C − A. The real skill lies in reversing the process. When a condition such as a required water depth is given, you substitute it into the model and solve for t, isolating the sine term, then applying the inverse sine to find the angle. Because arcsin returns only the principal value, this gives the first time the condition is met, which must then be converted from hours into hours and minutes. Understanding how amplitude, period, and phase interact is what turns a simple equation into a genuine description of a real-world cycle.