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
A ship is moored beside a floating pontoon in a harbour. Waves make the deck of the ship and the surface of the pontoon rise and fall.

The heights are modelled for a period of 60 seconds. At time $t$ seconds after the start of the period, the deck of the ship is $s$ metres above mean sea level and the surface of the pontoon is $p$ metres above mean sea level, where

$s = 4.2 - 0.35\sin t$

and

$p = 0.8 - 0.6\cos t$
(a) Show that, at the start of the period, the deck is 4 metres above the surface of the pontoon.

(1 mark)
(b) At time $t$ seconds the deck is $d$ metres above the surface of the pontoon.

Show that

$d = 3.4 + R\cos(t + \alpha)$

stating the values of the positive constants $R$ and $\alpha$

(5 marks)
(c) Hence find the least height of the deck above the surface of the pontoon during the 60 seconds, giving your answer to the nearest centimetre.

(2 marks)
Level 4 · 8 marksTry it
2
A child is playing on a swing in a park.

The seat of the swing moves backwards and forwards. In a model of this motion
- the seat is at its greatest height, $1.7$ m above the ground, at each end of a swing
- the seat is at its least height, $0.5$ m above the ground, as it passes through the lowest point of a swing
The height of the seat above the ground, $H$ $H$ m, $t$ $t$ seconds after the seat is released from one end of a swing, is modelled by the equation

$H = a + b(t - 1)^2$

where $a$ $a$ and $b$ $b$ are positive constants.
(a) Find a complete equation for the model.

(3 marks)
(b) Find, according to the model, the height of the seat above the ground when $t = 0.5$ $t = 0.5$

(1 mark)

A second model for the height of the seat above the ground, $H$ $H$ m, $t$ $t$ seconds after the seat is released, is

$H = 0.6\sin(180t + \alpha)^\circ + \beta$

where $\alpha$ $α$ and $\beta$ $β$ are constants and $0 \leqslant \alpha < 360$ $0 ⩽ α < 360$
(c) Find a complete equation for the second model.

(2 marks)

The child keeps swinging in the same way for $3$ $3$ minutes.
(d) Explain why the second model is more appropriate than the first model over this time.

(1 mark)
Level 4 · 7 marksTry it
3
A cyclist rides at a constant speed along a straight horizontal road. A small reflector is fixed to one of the spokes of the front wheel of the bicycle.

The height of the reflector above the road, $H$ $H$ cm, $t$ $t$ seconds after the cyclist passes a lamp post, is modelled by the equation

$H = \left|A\sin(bt + \alpha)^\circ\right|$

where $A$ $A$ , $b$ $b$ and $\alpha$ $α$ are constants.

In the model
- the reflector reaches its greatest height above the road, $58$ cm, once in each revolution of the wheel
- the wheel makes one complete revolution every $0.6$ seconds
- when $t = 0$ the reflector is $20$ cm above the road and is rising
(a) Find a complete equation for the model. Give $A$ $A$ and $b$ $b$ as exact values and give $\alpha$ $α$ correct to 3 significant figures.

(4 marks)

A second model for the height of the reflector above the road is

$H = \left|A\sin(bt + \alpha)^\circ\right| + d$

where $d$ $d$ is a positive constant.
(b) Give a reason why the second model is more realistic than the model in part (a).

(1 mark)
Level 4 · 5 marksTry it
4
(a) Express $8\cos\theta + 15\sin\theta$ in the form $R\cos(\theta - \alpha)$ , where $R > 0$ and $0 < \alpha < 90^\circ$

State the value of $R$ and give the value of $\alpha$ , in degrees, to 2 decimal places.

(3 marks)

The air temperature, $T\ ^\circ$ C, at a weather station in a desert, $t$ hours after midnight on a particular day, is modelled by the equation

$T = 15 - 8\cos(15t)^\circ - 15\sin(15t)^\circ \qquad 0 \leqslant t < 24$
(b) Using your answer to part (a), find

(i) the maximum temperature at the weather station on that day according to the model,

(ii) the time of day at which this maximum temperature occurs according to the model, giving your answer to the nearest minute.

(3 marks)

On that day, the lowest temperature recorded at the weather station was $4\ ^\circ$ C, at 06:30
(c) Use this information to comment on the model.

(2 marks)
Level 4 · 8 marksTry it
5
The number of hours of daylight, $D$ $D$ , in a town on the day that is $t$ $t$ days after 21 March is modelled by the equation

$D = A + B\sin(kt)^\circ$

where $A$ $A$ , $B$ $B$ and $k$ $k$ are positive constants.

Given that, according to the model,
- $D$ is periodic with period $365$ days
- the greatest number of hours of daylight in the town is $16.4$
- the least number of hours of daylight in the town is $7.6$
(a) find a complete equation for the model.

(3 marks)
(b) Write down the value of $t$ $t$ on which, according to the model, the town has the greatest number of hours of daylight.

(1 mark)
(c) Use the model to find the number of hours of daylight in the town when $t = 41$ $t = 41$ , giving your answer in hours and minutes to the nearest minute.

(2 marks)

On 21 December, when $t = 275$ $t = 275$ , the town actually had $7.9$ $7.9$ hours of daylight.
(d) Use this information to comment on the model.

(2 marks)
Level 4 · 8 marksTry it

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