A-level · Trigonometry

Modelling with trigonometryExam-style questions

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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10 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 4 Level 5

1
The depth of water in a harbour, $d$ metres, is modelled by

$d = 5 + 2\sin(30t)^{\circ}$

where $t$ is the time in hours after midnight.
(a) Write down the maximum depth, and find the first time at which it occurs. (2 marks)
(b) Find the first time at which the depth is $6$ metres. (2 marks)
(c) State the period of the model, and interpret it. (1 mark)
Level 5 · 5 marksTry it
2
(a) Given that

$56\cos\theta - 33\sin\theta \equiv K\cos(\theta + \alpha)$

where $K > 0$ and $0 < \alpha < 90^\circ$ , find the value of $K$ and the value of $\alpha$ , giving $\alpha$ in degrees to 2 decimal places.

(3 marks)

The manager of a shop records the number of umbrellas and the number of pairs of sunglasses that the shop sells each week over one year.

The number of umbrellas sold in a week, $U$ , is modelled by the equation

$U = A + 56\cos(30t)^\circ - 33\sin(30t)^\circ$

where $t$ is the time in months after the start of the year and $A$ is a constant.

According to the model, the shop sells at most $245$ umbrellas in a week.
(b) (i) Find a complete equation for the model.

(ii) Hence state, according to the model, the least number of umbrellas that the shop sells in a week.

(2 marks)

The manager's records show that sales of umbrellas were actually lowest at the start of August.
(c) In the light of this information, comment on the model for the number of umbrellas.

(2 marks)

The number of pairs of sunglasses sold in a week, $S$ , is modelled by the equation

$S = 95 + 75\sin(30t - 95)^\circ$

According to this model, sales of sunglasses are greatest when $t = T$
(d) Find, according to the models, the number of umbrellas sold in a week when $t = T$

(4 marks)
Level 5 · 11 marksTry it
3
(a) Write $21\sin\theta + 20\cos\theta$ in the form $R\sin(\theta + \alpha)$ , where $R > 0$ and $0 < \alpha < \dfrac{\pi}{2}$ , stating the exact value of $R$ and giving the value of $\alpha$ in radians to 3 decimal places.

(3 marks)

An environmental agency monitors a river at one location for a year.

The rate of flow of the river at this location, $F$ m $^3$ s $^{-1}$ , is modelled by the equation

$F = A + 21\sin\left(\frac{\pi t}{6}\right) + 20\cos\left(\frac{\pi t}{6}\right)$

where $t$ is the time in months after the start of the year and $A$ is a constant.

According to the model, the least rate of flow of the river at this location is $6$ m $^3$ s $^{-1}$
(b) (i) Find a complete equation for the model.

(ii) Hence state, according to the model, the greatest rate of flow of the river at this location.

(2 marks)

The agency's measurements show that the rate of flow was actually at its lowest in the middle of August.
(c) Explain what this information suggests about the model for the rate of flow.

(2 marks)

The temperature of the water at the same location, $W\ ^\circ$ C, is modelled by the equation

$W = 11 - 6\cos\left(\frac{\pi t}{6} - 0.5\right)$

According to this model, the water is coldest when $t = T$
(d) Find, according to the models, the rate of flow of the river when $t = T$ , giving your answer to 3 significant figures.

(4 marks)
Level 5 · 11 marksTry it