O-Level Trigonometric functions, identities and equations: worked solution

Question

At a harbour, the water depth d metres at t hours after 08 00 is described by the sinusoidal model d = 2 + 1.25 sin(πt/6). Draw a sketch of the graph of d against t across a 24-hour period.

d = 2 + 1.25 sin(πt/6). Midline d = 2, amplitude 1.25 (so max d = 3.25, min d = 0.75). Period = 2π / (π/6) = 12 hours, so two complete cycles across 0 ≤ t ≤ 24. Start at t = 0: d = 2, rising. Maximum 3.25 at t = 3; back to 2 at t = 6; minimum 0.75 at t = 9; back to 2 at t = 12; the pattern repeats to t = 24. Sketch a smooth sine wave oscillating between 0.75 and 3.25 about d = 2, with this shape.