A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 7: Trigonometry and modelling · Lesson 6

Modelling with trigonometric functions

117 marks

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Date

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Quick questions to start.

A1
Last lesson
Write $20 cos x + 21 sin x$ in the form $R cos (x - α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ . Give $α$ to 2 decimal places.
[3 marks]
Answer
A2
Today's idea
The depth of water in a harbour is $D = 5 + 3 sin (30 t)^{\circ}$ metres. State the greatest depth and the least depth.
[2 marks]
Answer
A3
Needed today
$t$ is in hours. State the period of $sin (30 t)^{\circ}$ and the period of $7 - 2 cos (15 t)^{\circ}$ .
[2 marks]
Answer
A4
From chapter 6
Solve $sin (30 t)^{\circ} = 0.5$ for $0 \leq t < 12$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

The height above the ground of a car on a big wheel is $H = 18 - 16 cos (12 t)^{\circ}$ metres, $t$ minutes after the car is at its lowest point. Find the greatest height, the first time it happens, and the time for one turn of the wheel.

Hint −16 cos(12t)° is greatest when cos(12t)° = −1. One turn is when 12t goes up by 360.

B1
A car on a different wheel has height $H = 20 - 18 cos (9 t)^{\circ}$ metres, $t$ minutes after it is at its lowest point. Find the greatest height and the first time it happens, the time for one turn, and the first time the car is $29$ m above the ground.
[5 marks]
Answer

Example 2

The temperature in a town is modelled by $T = 12 + 3 cos (15 t)^{\circ} + 4 sin (15 t)^{\circ}$ °C, $t$ hours after midnight. Find the greatest temperature and the first time it happens, to the nearest minute.

Hint Write 3 cos(15t)° + 4 sin(15t)° as R cos(15t − α)° first. Then the greatest value of the cosine is 1.

B2
The temperature in a greenhouse is modelled by $T = 15 + 12 sin (15 t)^{\circ} - 5 cos (15 t)^{\circ}$ °C, $t$ hours after midnight. Write $12 sin (15 t)^{\circ} - 5 cos (15 t)^{\circ}$ in the form $R sin (15 t - α)^{\circ}$ , where $R > 0$ and $0 < α < 90$ . Hence find the greatest temperature and the first time it happens, to the nearest minute.
[6 marks]
Answer

Example 3

The depth of water at a harbour entrance is $D = 6 + 2.5 sin (30 t)^{\circ}$ metres, $t$ hours after midnight, $0 \leq t < 12$ . A boat needs at least $7$ m of water. Find the times between which the boat can enter, to the nearest minute.

Hint Solve the equality first: 6 + 2.5 sin(30t)° = 7. The depth is at least 7 m between the two solutions.

B3
The depth of water in a harbour is $D = 7 - 3 cos (30 t)^{\circ}$ metres, $t$ hours after midnight, $0 \leq t \leq 12$ . A ship needs at least $8$ m of water. Find the times between which the ship can enter, to the nearest minute.
[5 marks]
Answer

Example 4

On one day high tide is $8.2$ m at $03 : 00$ and the next low tide is $1.4$ m at $09 : 00$ . The depth is modelled by $D = a + b sin (k t)^{\circ}$ , $t$ hours after midnight. Find $a$ , $b$ and $k$ , and give one reason the model may not work on later days.

Hint a is halfway between the greatest and least depths, b is half the difference, and high to low is half a period.

B4
The temperature in a greenhouse is modelled by $T = p - q cos (15 t)^{\circ}$ °C, $t$ hours after midnight, where $p$ and $q$ are positive. The least temperature is $9$ °C at midnight and the greatest is $25$ °C at noon. Find $p$ and $q$ , find the temperature the model gives at $4$ pm, and give one reason the model may not suit a different day.
[5 marks]
Answer

Practice

A-Level

C1
The height of a seat on a big wheel is $H = 25 - 22 cos (10 t)^{\circ}$ metres, $t$ minutes after it is at its lowest point. Kim says: "The wheel turns once every $10$ minutes, because the number in front of $t$ is $10$ ." Explain Kim's mistake. Find how long, in each turn, the seat is more than $36$ m above the ground.
[4 marks]
Answer

C2
The depth of water in a harbour is $D = 10 + 3 sin (30 t)^{\circ} - 4 cos (30 t)^{\circ}$ metres, $t$ hours after midnight. Write $3 sin (30 t)^{\circ} - 4 cos (30 t)^{\circ}$ in the form $R sin (30 t - α)^{\circ}$ , where $R > 0$ and $0 < α < 90$ . Hence find the greatest depth and the first time after midnight it happens, to the nearest minute.
[5 marks]
Answer
C3
A weight on a spring moves so that its displacement from a fixed point $O$ is $x = 3 cos 2 t + 4 sin 2 t$ cm, $t$ seconds after it is released. The angle $2 t$ is in radians. Find the greatest distance of the weight from $O$ , and the first time, $t > 0$ , when $x = - 2.5$ .
[5 marks]
Answer
C4
The number of hours of daylight in a city is modelled by $H = 12 - 4 cos (\frac{360 d}{365})^{\circ}$ , where $d$ is the number of days after $1$ January. Find the greatest number of hours of daylight and the value of $d$ where it happens. Find how many days of the year the model gives more than $14$ hours of daylight.
[5 marks]
Answer
C5
On one day high tide is $5.6$ m at $01 : 30$ and the next low tide is $1.2$ m at $07 : 45$ . The depth is modelled by $D = a + b cos (k (t - 1.5))^{\circ}$ metres, $t$ hours after midnight. Find $a$ , $b$ and $k$ , and the depth the model gives at $05 : 00$ , to 2 decimal places.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The height of a passenger on a big wheel is modelled by $h = 16 - 7 sin (6 t)^{\circ} - 73 cos (6 t)^{\circ}$ metres, $t$ minutes after the ride starts.
(a)
Write $7 sin (6 t)^{\circ} + 73 cos (6 t)^{\circ}$ in the form $R sin (6 t + α)^{\circ}$ , where $R > 0$ and $0 < α < 90$ .
[3 marks]
Answer
(b)
Find the greatest height of the passenger and the first time it happens.
[3 marks]
Answer
(c)
Find for how long, in each turn of the wheel, the passenger is more than $23$ m above the ground.
[3 marks]
Answer
(d)
Find the exact height of the passenger when the ride starts, and the least height of the passenger.
[2 marks]
Answer
Total for D1: 11 marks

D2
Ravi says: "So the depth is more than $5$ m only from midnight until $02 : 13$ ."
The depth of water in a harbour is modelled by $D = 4 + 2.5 cos (30 t)^{\circ}$ metres, $t$ hours after midnight. Ravi wants the times when the depth is more than $5$ m. He solves $4 + 2.5 cos (30 t)^{\circ} = 5$ , gets $cos (30 t)^{\circ} = 0.4$ , so $30 t = 66.42$ and $t = 2.214$ .
(a)
Explain why Ravi is wrong, and find all the times between midnight and noon when the depth is more than $5$ m, to the nearest minute.
[4 marks]
Answer
(b)
Find the total time between $00 : 00$ and $24 : 00$ when the model gives a depth of more than $5$ m. Give your answer in hours and minutes.
[3 marks]
Answer
(c)
Give two reasons why the model may not give the depth well on a later day.
[2 marks]
Total for D2: 9 marks
D3
The temperature in an office is modelled by $θ = 19 + 23 cos (15 t)^{\circ} + 2 sin (15 t)^{\circ}$ °C, $t$ hours after $06 : 00$ , $0 \leq t < 24$ . The air conditioning runs whenever $θ > 21$ . Find the number of hours each day the air conditioning runs, and the time it first switches off.
[6 marks]
Answer
D4
A buoy moves up and down on the sea. Its height above its rest level is modelled by $y = 0.6 cos 2 t - 0.8 sin 2 t$ metres, $t$ seconds after a wave passes, $t \geq 0$ . The angle $2 t$ is in radians.
(a)
Write $y$ in the form $R cos (2 t + α)$ , where $R > 0$ and $0 < α < \frac{π}{2}$ . Give $α$ to 4 decimal places.
[3 marks]
Answer
(b)
Find the greatest height of the buoy above its rest level and the smallest value of $t$ at which it happens.
[3 marks]
Answer
(c)
Find the values of $t$ in $0 \leq t \leq π$ when the buoy is $0.5$ m above its rest level. Give your answers to 3 significant figures.
[4 marks]
Answer
Total for D4: 10 marks
D5
On one day high tide at a port is $9.8$ m at $04 : 30$ and the next low tide is $2.2$ m at $10 : 30$ . The depth is modelled by $D = a + b cos (k (t - 4.5))^{\circ}$ metres, $t$ hours after midnight.
(a)
Find $a$ , $b$ and $k$ .
[3 marks]
Answer
(b)
A boat in the port needs at least $4$ m of water. Find the latest time after $04 : 30$ at which it can leave, to the nearest minute.
[3 marks]
Answer
(c)
The next high tide is at $16 : 55$ , not at the time the model gives. Use this to find a better value of $k$ , to 3 significant figures.
[2 marks]
Answer
Total for D5: 8 marks
D6
A big wheel has radius $30$ m and its centre is $33$ m above the ground. It turns at a steady speed once every $20$ minutes. A passenger $P$ is at the lowest point when $t = 0$ , where $t$ is in minutes. $h$ m is the height of $P$ above the ground.
(a)
Show that $h = 33 - 30 cos (18 t)^{\circ}$ .
[3 marks]
(b)
Find the first time $P$ is $48$ m above the ground. Give your answer in minutes and seconds.
[3 marks]
Answer
(c)
A second passenger $Q$ is at the lowest point when $t = 5$ . Find the first time after $t = 5$ when $P$ and $Q$ are at the same height, and that height to 3 significant figures.
[4 marks]
Answer
Total for D6: 10 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Two cars are on different big wheels. Their heights above the ground are $h_{1} = 20 - 18 cos (12 t)^{\circ}$ metres and $h_{2} = 15 - 12 sin (12 t)^{\circ}$ metres, $t$ minutes after they start, $0 \leq t < 30$ . Find the times when the two cars are at the same height, and the greatest vertical distance between the two cars. Give your answers to 3 significant figures.
[9 marks]
Hint 1 · What to try
Set $h_{1} = h_{2}$ and collect the cosine and sine terms on one side.
Hint 2 · The first line
You should get $18 cos (12 t)^{\circ} - 12 sin (12 t)^{\circ} = 5$ . Write the left side as $R cos (12 t + α)^{\circ}$ .
Hint 3 · The full method
$h_{1} - h_{2} = 5 - R cos (12 t + α)^{\circ}$ . It is furthest from $0$ when the cosine is $- 1$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can read the greatest value, least value and period from a model such as a + b cos(kt)°.
I can find every time in an interval when a model takes a given value, and how long it stays above it.
I can write a model with a sine and a cosine as one wave to find its greatest value and when it happens.
I can find a, b and k from data and give a reason a model may not fit.

Answers and mark scheme

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