Trigonometry · step 13 of 13

Modelling periodic phenomena with sinusoids

Any regular cycle can be modelled by y = a sin(b(t − h)) + k, where k is the midline, a the amplitude, 2π/b the period and h the horizontal shift. Building the model is a fixed recipe: amplitude is half the range, the midline is the average of the maximum and minimum, and b comes from the observed period.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Find the midline kk = (max + min)/2 — the value the quantity oscillates about.
Find the amplitude aa = (max − min)/2, always positive.
Find b from the periodb = 2π/period, using the same time units throughout.
Find the shift h from a known featureA sine model reaches its maximum a quarter period after t = h. Use a stated maximum time to solve for h.

Worked example

A tide has high water 5.6 m and low water 1.2 m, with high tide at 3:00 and a 12-hour period. Model the depth.

Midline: k = (5.6 + 1.2)/2 = 3.4. Amplitude: a = (5.6 − 1.2)/2 = 2.2.
b = 2π/12 = π/6.
A sine peaks a quarter period after the shift, so h = 3 − 3 = 0 hours.
So d(t) = 2.2 sin(π(t)/6) + 3.4 — check t = 3: 2.2 sin(π/2) + 3.4 = 5.6 ✓.

Answer. d(t) = 2.2 sin(πt/6) + 3.4 metres, with t in hours after midnight.

Where marks get dropped

These are the specific errors that cost credit on modelling periodic phenomena with sinusoids questions — QED's rubric penalises each of them separately.

Using the maximum as the amplitude. The amplitude is HALF the range, measured from the midline.
Mixing time units — computing b from a period in hours but evaluating with t in minutes.
Forgetting that a sine peaks a quarter period after its shift, while a cosine peaks AT its shift. Choosing cosine often removes the shift entirely.

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Modelling periodic phenomena with sinusoids — frequently asked questions

Should I use sine or cosine?

Whichever needs less shifting. If you know the time of a maximum, cosine is natural since it peaks at t = h; if you know a midline crossing, sine is easier.

How do I find when the quantity exceeds a level?

Solve the inequality by first solving the equation, then using the graph’s symmetry to identify the interval between consecutive solutions.

What if the data are not perfectly periodic?

A sinusoid is then an approximation. Real tides need several superposed components, which is what harmonic analysis provides.

The rest of Trigonometry

Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.

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