Trig models

The idea

A model such as $D = a + b\cos(kt)^\circ$ swings between $a - b$ and $a + b$ , and repeats every $\frac{360}{k}$ units of time. Check that the model's values and times make sense in the situation.

Maximum $a + |b|$ , minimum $a - |b|$ .
Period $= \frac{360^\circ}{k}$ (or $\frac{2\pi}{k}$ in radians).
To find a time, solve the trig equation and keep the solutions in the allowed range.

Watch out A model is only valid over the times it was built for.

See it worked

Worked example

The depth of water in a harbour is $D = 5 + 2\sin(30t)^\circ$ metres, $t$ hours after midnight. Find the greatest depth and when it first happens.

Greatest depth when $\sin(30t)^\circ = 1$ : $D = 7\text{ m}$ .
$30t = 90$ .
$t = 3$ : at 3 am.

Answer $7\text{ m}$ at 3 am

Your turn

Question 1Non-calculator3 marks

The depth of water is $D = 6 + 3\cos(30t)^\circ$ metres. Find the greatest depth.

Question 2Non-calculator3 marks

A height is modelled by $h = 4\sin(15t)^\circ$ , with $t$ in hours. Find the period of the model.

Question 3Non-calculator3 marks

Using $D = 6 + 3\cos(30t)^\circ$ , find the first time $t > 0$ when the depth is $7.5\text{ m}$ .

In past papers

13 official questions in 12 papers since 2018. The latest:

All A-level Maths past papers

A-level Maths Pro, Pearson Edexcel 9MA0

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