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Trigonometric modelling questions
Quantities that repeat, such as tides, daylight hours and the height of a point on a wheel, are modelled with sine and cosine. This lesson covers reading such a model, identifying the centre line, amplitude, period and phase, and using the harmonic form on models that carry two trigonometric terms at once.
7 original questions · 24 marks · the trigonometric modelling notes · Trigonometry
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.
Write down the amplitude and period of y = 3 sin 2t, with t in seconds.
Worked answer
The amplitude is 3 and the period is 2π/2 = π seconds. B1 for the amplitude, B1 for the period. The 2 inside the bracket speeds the wave up and lives in the period. The 3 outside scales the swing and lives in the amplitude. Quoting the period as 2 is the reflex to resist; the coefficient divides into 2π rather than being the answer.A model gives h = 5 + 2 cos t. Write down the maximum and minimum values of h.
Worked answer
The cosine swings between ±1, so h runs from 5 − 2 = 3 up to 5 + 2 = 7. B1 for the maximum, B1 for the minimum. The 5 sets the centre line and the 2 the swing about it. Both values are wanted, and the maximum on its own scores half.A wheel's seat height is modelled by h = 10 − 8 cos (πt/20), in metres and seconds. Write down the height of the axle and the radius of the wheel, and find the maximum seat height and the period of the ride.
Worked answer
The axle sits on the centre line, 10 m up, and the radius is the amplitude, 8 m. The seat tops out at 10 + 8 = 18 m and bottoms at 2 m. The period is 2π ÷ (π/20) = 40 s. B1 for the axle height, B1 for the radius, B1 for the maximum height, B1 for the period. The minus sign starts the seat at the bottom of the wheel at t = 0, where a ride starts.A wheel's seat height is modelled by h = 10 − 8 cos (πt/20), in metres and seconds. Find the first time the seat is 14 m high.
Worked answer
Setting 10 − 8 cos (πt/20) = 14 gives cos (πt/20) = −½. The first positive angle with that cosine is 2π/3, so πt/20 = 2π/3 and t = 40/3 = 13.3 s to 3 significant figures. M1 for cos (πt/20) = −½, M1 for πt/20 = 2π/3, A1 for 13.3 s, B1 for taking the first crossing. Later crossings exist at 4π/3 and beyond, but the question asks for the first. Take the principal value, then translate back through the coefficient, and keep the calculator in radians throughout.A buoy's height above mean sea level is y = 0.9 sin 0.6t, in metres and seconds. Find the amplitude, the period, and the first positive time at which the buoy is at mean sea level.
Worked answer
The amplitude is 0.9 m and the period is 2π/0.6 = 10.5 s. Mean sea level means y = 0. The buoy starts there at t = 0, and the sine next vanishes when 0.6t = π, so t = 5.24 s, half a period later. B1 for the amplitude, B1 for the period, M1 for 0.6t = π, A1 for 5.24 s. Reading the first positive time as 0 is the trap the word positive is there to close.A tide gauge fits the data y = 2 sin 0.5t + 1.5 cos 0.5t, in metres and hours. Express y as a single sine wave, and find the height of high tide and the first time it occurs.
Worked answer
R = √(4 + 2.25) = 2.5 and tan α = 1.5/2 gives α = 0.6435, so y = 2.5 sin (0.5t + 0.6435). High tide is 2.5 m, where the sine reaches 1: 0.5t + 0.6435 = π/2, so t = 1.85 hours. M1 for the harmonic form, A1 for R = 2.5, M1 for tan α, A1 for α = 0.6435, B1 for the high tide, A1 for t = 1.85 hours. Two terms, one tide: the harmonic form is the tool that makes the data readable.A student claims that doubling the amplitude of a periodic model makes it oscillate twice as fast. Untangle the claim.
Worked answer
Amplitude and period are separate dials. Doubling the amplitude doubles how far the oscillation swings and leaves its timing alone; oscillating faster needs a bigger coefficient inside the bracket, which shortens the period. B1 for amplitude and period being independent, B1 for what does change the speed. The claim muddles the outside dial with the inside one.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise trigonometric modelling one question at a time
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