Pure · Trigonometry · A-level

Modelling with trigonometry

Using a sine or cosine model for something that repeats — a tide, a day's temperature, the hours of daylight, a turning wheel. What the midline, amplitude, period and shift mean in context; reading greatest and least values; building a complete equation from a description; finding every time a value is reached, to the nearest minute; using the R-form when the model is a sum of two waves; and saying where a model breaks down.

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Why it works

A tide in four numbers

Anything that repeats on a cycle rises and falls like a sine wave: a tide, a day's temperature, the hours of daylight, a turning wheel. Here is the model this page uses throughout. The depth of water at the end of a pier, DD metres, tt hours after midnight, is modelled by

D=6+2.5sin⁡(30(t−1))∘,0≤t<24.D = 6 + 2.5\sin\big(30(t - 1)\big)^\circ, \qquad 0 \le t < 24.

Each number is a dial with a meaning in the harbour. The 66 is the midline, the level the water rises above and falls below. The 2.52.5 is the amplitude, how far above and below that level it goes. The 3030 sets the period: the bracket gains 30∘30^\circ an hour, so a full turn takes 1212 hours. The 11 is the shift: a sine is on its midline and rising when its bracket is 0∘0^\circ, so at 01:00 the water is 66 m deep and coming in. High water follows when the bracket reaches 90∘90^\circ, at t=4t = 4.

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The rest of the explanation, plus 3 worked examples you step through move by move.

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