Modelling with Trigonometric Functions (AQA A Level Maths: Pure): Exam Questions

The alternating voltage, $V$ volts, in an electrical circuit, $t$ seconds after it is switched on, is modelled by the equation

$V=55\sqrt{3} \text{sin}\left(\frac{\pi t}{30}\right)+55\text{cos}\left(\frac{\pi t}{30}\right)t\ge 0$

Show that

$55\sqrt{3} \text{sin}\left(\frac{\pi t}{30}\right)+55\text{cos}\left(\frac{\pi t}{30}\right)$

can be written in the form

$R \text{sin}\left(\frac{\pi t}{30}+\alpha \right)$

where $R=110$ and $\alpha =\frac{\pi }{6}$.

According to the model,

(i) find the voltage at the instant the circuit is switched on,

(ii) find the voltage exactly one minute after the circuit is switched on.

Find the first positive value of $t$ for which the voltage equals $-55$ volts.

A small spring is extended to its maximum length and released from rest. The length of the spring, $l$ cm, $t$ seconds after it is released, is modelled by the equation

$l=5+3\text{cos}2tt\ge 0$

According to the model,

(i) write down the natural length of the spring,

(ii) write down the maximum extension of the spring.

(i) Find the length of the spring exactly $6$ seconds after it is released.

(ii) Find the first positive value of $t$ for which the length of the spring is exactly $4$ cm.

State one limitation of the model for large values of $t$.

A dolphin is diving in and out of the water at a constant speed. The height, $h$ m, of the dolphin relative to sea level, $t$ seconds after it begins a jump from sea level, is modelled by the equation

$h=A\text{sin}\left(Bt\right)t\ge 0$

where $A$ and $B$ are constants.

According to the model,

• on each jump the dolphin reaches a maximum height of exactly $2$ m above sea level

• on each dive the dolphin reaches a maximum depth of exactly $2$ m below sea level

• starting from sea level, the dolphin takes exactly $\frac{2\pi }{3}$ seconds to jump out of the water, dive back under and return to sea level

• $0<B\le 5$.

Find the values of $A$ and $B$.

Figure 1 shows a sketch of the path of a swing boat fairground ride.

The path of the swing boat is modelled as a semi-circle of radius exactly $8$ m. The height of the swing boat above ground level, $y$ m, at a horizontal displacement of $x$ m from a fixed origin $O$ on the ground is modelled by the equation

$y=10-\sqrt{64-x^2}-8\le x\le 8$

The initial position of the swing boat is at the point $(0,2)$.

According to the model,

(i) find the height of the swing boat when its horizontal displacement from $O$ is exactly $2$ m,

(ii) find the exact horizontal distance of the swing boat from $O$ when its height is exactly $6$ m.

The horizontal displacement of the swing boat, $x$ m, $t$ seconds after it is released from its initial position is modelled by the equation

$x=8\text{sin}\left(\frac{\pi t}{6}\right)t\ge 0$

Find the time it takes the swing boat to swing from one end of the ride to the other.

The height of water in a reservoir, $h$ m, $t$ hours after midnight is modelled by the equation

$h=A+B\text{sin}\left(\frac{\pi t}{6}\right)t\ge 0$

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

According to the model, write down, in terms of $A$ and $B$,

(i) the natural height of the water in the reservoir,

(ii) the maximum height of the water,

(iii) the minimum height of the water.

Given also that

• the maximum level of the water is exactly $3$ m higher than its natural level

• the maximum level of the water is exactly three times its minimum level,

find the maximum, minimum and natural water levels.

(i) Deduce the number of times per day the water reaches its maximum level.

(ii) Find the times of day when the water level is at its minimum.

Figure 1 shows a sketch of a Ferris wheel.

A Ferris wheel is modelled as a circle in the $x$–$y$ plane with centre $(0,0)$ and radius exactly $50$ m.

According to the model,

• there are exactly $p$ passenger pods evenly spaced around the Ferris wheel

• the position of a passenger pod can be determined by the angle $\theta$ radians, measured anticlockwise from the positive $x$-axis

• the coordinates $(x,y)$ of a passenger pod are given by

$(x,y)=(A\text{cos}\theta ,A\text{sin}\theta )$

where $A$ is a positive constant

• ground level is represented by the horizontal line with equation $y=-60$.

(i) Write down the value of $A$.

(ii) Given that the exact angle between adjacent passenger pods is $\frac{\pi }{12}$ radians, find the value of $p$.

(iii) Find the maximum height of a passenger pod above the ground.

Find, to $3$ significant figures, the value of $\theta$ for a passenger pod located at the point $(30,40)$.

A lifejacket falls over the side of a boat from a height of exactly $3$ m above sea level. The height of the lifejacket relative to sea level, $h$ m, $t$ seconds after it falls, is modelled by the equation

$h=3{\text{e}}^{-0.7t}\text{cos}4tt\ge 0$

Given that, according to the model, the lifejacket reaches its furthest point below sea level exactly $0.742$ seconds after it falls, find the total distance the lifejacket has fallen at this time, giving your answer to $3$ significant figures.

Find the exact values of $t$ for the first three times the lifejacket is at sea level.

(i) Find the value of $3{\text{e}}^{-0.7t}$ when $t=6.2$.

(ii) Hence justify why, for all $t\ge 6.2$, the lifejacket will always remain within exactly $4$ cm of sea level.

The number of daylight hours in the UK, $h$, $d$ days after the spring equinox is modelled by the equation

$h=12+\frac{9}{2}\text{sin}\left(\frac{2\pi d}{365}\right)d\ge 0$

where the spring equinox is defined as the day in spring when the number of daylight hours is exactly $12$.

According to the model,

(i) find the number of daylight hours exactly $100$ days after the spring equinox,

(ii) find the two values of $d$ for which the number of daylight hours is exactly $9$, giving your answers to the nearest integer.

Find the number of days of the year for which the number of daylight hours exceeds exactly $15$ hours, giving your answer as a whole number of days.

The length of a spring, $l$ cm, $t$ seconds after being released from rest is modelled by the equation

$l=a+b\text{cos}4tt\ge 0$

where $a$ and $b$ are constants.

According to the model, state what the constants $a$ and $b$ represent in terms of the length of the spring.

Given that the minimum length of the spring is exactly $12$ cm and the maximum length of the spring is exactly $30$ cm, find the value of $a$ and the value of $b$.

The length of a similar spring, with the same values of $a$ and $b$, is modelled by the equation

$l=a+b\text{cos}2tt\ge 0$

(i) Compare the motion of this spring with the motion of the original spring.

(ii) State one refinement to the models that would make them more realistic.

The height of a hovering helicopter above the ground, $h$ metres, at time $t$ seconds is modelled by the equation

$h=A+B \text{cos} Ct,t\ge 0$

where $A$, $B$ and $C$ are constants.

Given that, according to the model,

• the helicopter moves between a minimum height of exactly $200$ m and a maximum height of exactly $220$ m

• it takes exactly $\frac{\pi }{5}$ seconds for the helicopter to move between these two heights

find a complete equation for the model, giving the values of $A$, $B$ and $C$.

State the initial height of the helicopter suggested by the model.

Figure 1 shows a sketch of the path of a swing boat fairground ride, which swings forwards and backwards.

The path of the swing boat is modelled as a semicircle of radius exactly $10$ metres.

According to the model, at time $t$ seconds,

• the horizontal displacement of the swing boat from a fixed origin $O$ is $x$ metres

• the height of the swing boat above the ground is $h$ metres

where $x$ and $h$ are modelled by the equations

$x=10 \text{sin}\left(\frac{\pi t}{5}\right),t\ge 0$

$h=12-10\left|\text{cos}\left(\frac{\pi t}{5}\right)\right|,t\ge 0$

Verify that the initial position of the swing boat is $(0,2)$.

According to the model,

(i) write down the coordinates of the two points at which the swing boat is at its maximum height,

(ii) hence find the time it takes the swing boat to swing between these two points.

(i) Find the exact position of the swing boat when it has swung through an angle of exactly $\frac{\pi }{6}$ radians anticlockwise from the $y$-axis, as shown in Figure 1.

(ii) Hence find the first positive value of $t$ at which the swing boat reaches this position.

The height of water in a reservoir, $h$ metres, $t$ hours after midnight is modelled by the equation

$h=A+B \text{sin} Ct,t\ge 0$

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

Given that, according to the model,

• the water level rises and falls through exactly one and a half cycles in a 24-hour period

• the minimum height of the water is exactly $1$ m, which occurs just once per day

• the maximum height of the water is exactly $11$ m

(i) find the exact value of $C$,

(ii) find the time of day when the minimum height of the water occurs,

(iii) find the value of $A$ and the value of $B$.

The reservoir can hold a maximum of exactly $10$ m of water. If this height is exceeded, an overflow reservoir is used.

Find the times of day during which the overflow reservoir is in use, giving your answers to the nearest minute.

Figure 1 shows a sketch of a Ferris wheel.

The Ferris wheel is modelled as a circle in the $x$-$y$ plane, with centre $(0,0)$ and radius exactly $60$ m.

According to the model,

• there are exactly $30$ passenger pods evenly spaced around the Ferris wheel

• the position of a passenger pod can be determined by the angle $\theta$ radians, measured anticlockwise from the positive $x$-axis, as shown in Figure 1

• the coordinates $(x,y)$ of a passenger pod are given by

$(x,y)=\left(A \text{cos} \theta ,B \text{sin} \theta \right)$

where $A$ and $B$ are positive constants

• ground level is represented by the horizontal line with equation $y=-62$

(i) Write down the value of $A$ and the value of $B$.

(ii) Find the exact angle, in radians, between adjacent passenger pods.

Find the height above the ground of a passenger pod when $\theta =\frac{7\pi }{6}$.

Find, to 3 significant figures, the value of $\theta$ for a passenger pod located at the point $(48,-36)$.

Evaluate the model in the case where $A\ne B$, giving a reason for your answer.

A lifejacket falls from a ship from an initial height of exactly $4$ m above sea level.

The height of the lifejacket relative to sea level, $h$ metres, $t$ seconds after it begins to fall is modelled by the equation

$h=A{e}^{-kt} \text{cos} 2t,t\ge 0$

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

According to the model,

(i) write down the value of $A$,

(ii) describe the effect of the constant $k$ on the height of the lifejacket as $t$ increases.

Given that the lifejacket is exactly $1$ m below sea level exactly $2.054$ seconds after it begins to fall,

find the value of $k$, giving your answer to one decimal place.

The number of daylight hours in the UK, $h$, $d$ days after the spring equinox, is modelled by the equation

$h=A+B \text{sin}\left(\frac{2\pi }{365}d\right),d\ge 0$

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

Given that, according to the model,

• on the spring equinox ($d=0$) there are exactly $12$ daylight hours

• the maximum number of daylight hours in the UK is exactly $16.5$

(i) write down the value of $A$,

(ii) write down the value of $B$.

Find the number of days of the year for which the number of daylight hours is less than exactly $10$, giving your answer as a whole number of days.

Given that the spring equinox occurs on 21 March, find the two dates of the year on which there are exactly $16$ daylight hours according to the model.

State one reason why the model would need to be adjusted every 4 years.

The alternating voltage, $V$ volts, in an electrical circuit, $t$ seconds after it is switched on, is modelled by the equation

$V=55\sqrt{2} \text{sin}\left(\frac{\pi t}{60}\right)+55\sqrt{2} \text{cos}\left(\frac{\pi t}{60}\right),t\ge 0$

Express

$55\sqrt{2} \text{sin}\left(\frac{\pi t}{60}\right)+55\sqrt{2} \text{cos}\left(\frac{\pi t}{60}\right)$

in the form

$R \text{sin}\left(\frac{\pi t}{60}+\alpha \right)$

where $R>0$ and $0<\alpha <\frac{\pi }{2}$

Give the exact value of $R$ and the exact value of $\alpha$ in radians.

According to the model,

(i) find the exact voltage at the instant the circuit is switched on,

(ii) write down the maximum voltage in the electrical circuit.

(i) Find the first positive value of $t$ for which the maximum voltage occurs.

(ii) Deduce the time it takes the voltage to complete one full period.

The length of a spring, $l$ cm, $t$ seconds after being released from rest, is modelled by the equation

$l=a+b \text{cos} ct,t\ge 0$

where $a$, $b$ and $c$ are positive constants.

According to the model,

(i) describe the effect of the constant $c$ on the motion of the spring,

(ii) explain how the equation shows that the spring is released from its maximum length.

Given that it takes exactly $\frac{\pi }{10}$ seconds from release until the spring first returns to its initial length,

(i) find the exact value of $c$.

Given also that the maximum length of the spring is exactly twice its minimum length,

(ii) find an equation linking $a$ and $b$.

State one reason why the model would not be appropriate for the length of the spring if $b\ge a$.

The height above the ground, $h$ metres, of a drone used as part of an air display, $t$ seconds after launch, is modelled by the equation

$h=A+B \text{sin} (Ct+D),t\ge 0$

where $A$, $B$, $C$ and $D$ are positive constants and $0<D<\frac{\pi }{2}$

Given that, according to the model,

• the drone is launched upwards from a height of exactly $23$ m

• the drone reaches its maximum height of exactly $26$ m exactly $\frac{\pi }{6}$ seconds after launch

• the minimum height of the drone is exactly $14$ m

find the value of $A$, the value of $B$, the value of $C$ and the exact value of $D$.

The drone's lights switch off when its height drops below exactly $17$ m.

Show that the drone's lights are on for exactly two thirds of each complete cycle.

Figure 1 shows a sketch of the path of a swing boat fairground ride.

The path of the swing boat is modelled as moving forwards and backwards along the arc of a semicircle in the $x$-$y$ plane.

According to the model,

• the semicircle has a radius of exactly $18$ m and its centre is at $(0,20)$

• the lowest point of the arc is at $(0,2)$

• the angle $\theta$ radians is the angle made with the downward vertical from the centre of the semicircle

Show that, for $0\le \theta \le \frac{\pi }{2}$,

(i) the $x$-coordinate of the swing boat is given by $x=18 \text{sin} \theta$,

(ii) the $y$-coordinate of the swing boat is given by $y=20-18 \text{cos} \theta$.

The model is refined so that the horizontal displacement from the origin, $x$ metres, and the height above the ground, $y$ metres, $t$ seconds after the swing boat is set in motion are given by the equations

$x=18 \text{sin} Bt,t\ge 0$

$y=20-18\left|\text{cos} Bt\right|,t\ge 0$

where $B$ is a positive constant.

(i) State one reason why the modulus of the cosine function is required for the $y$-coordinate.

Given that the time taken for the swing boat to swing between the two points of maximum height at either end of the ride is exactly $8$ seconds,

(ii) find the exact value of $B$.

Find the values of $t$, in the interval $0\le t\le 4$, for which the height of the swing boat above the ground is equal to its horizontal distance from the origin.

The height of water, $h$ m, in a reservoir is modelled by the equation

$h=A+B \text{sin} Ct,t\ge 0$

where $t$ is the time in hours after midnight, and $A$, $B$ and $C$ are positive constants.

Explain briefly what each of the constants $A$, $B$ and $C$ represents in the context of the model.

Show that the height of water will first be at its minimum level at time

$t=\frac{3\pi }{2C}$

hours after midnight.

Show that the magnitude of the rate of change of the height of water in the reservoir is at its maximum every

$\frac{k\pi }{C}$

hours after midnight, where $k$ is a non-negative integer.

Figure 1 shows a sketch of the design for a Ferris wheel.

The Ferris wheel is modelled as a circle in the $x$-$y$ plane with centre $(0,a)$ and radius $r$ metres.

According to the model,

• the $x$-axis represents ground level

• two symmetrical ground supports, shown as thick lines in Figure 1, each go from the centre of the Ferris wheel to ground level

• the left-hand ground support is represented by the equation $4x-3y+240=0$

• one of the passenger pods is located at the point with coordinates $(42,136)$

(i) Find the equation of the circle.

(ii) Hence find the distance from the ground to the lowest point of the Ferris wheel.

According to the design, exactly $p$ passenger pods are to be evenly distributed around the Ferris wheel. The engineers require that no more than $3$ pods are located within the angle formed by the two ground supports at any one time.

Find the maximum value of $p$ that this design approach allows.

For both strength and aesthetic reasons, each ground support will be made in two sections. Thinner material will be used for the section within the wheel, so as not to obstruct the view of, and from, the Ferris wheel, and thicker material will be used for the lower base section outside the wheel.

Find the percentage of each ground support that will be made from the thicker material.

The height of a helicopter, $h$ m, $t$ seconds after its engine is started, is modelled by the equation

$h=12+2 \text{tan}\left(\frac{t}{2}-\frac{\pi }{2}\right),0<t\le 6$

According to the model, there is a time lag before the helicopter takes off, which is represented by $h<0$ for the interval $0<t\le \alpha$.

Find the value of $\alpha$ to $2$ significant figures.

Show that the helicopter rises exactly $4$ m between $t=\frac{\pi }{2}$ seconds and $t=\frac{3\pi }{2}$ seconds.

Find the height of the helicopter at the instant the model ceases to be valid, giving your answer to the nearest metre.

The number of daylight hours, $h$, in the UK, $d$ days after the spring equinox (the day in spring when the number of daylight hours is exactly $12$) is modelled by the equation

$h=12+B \text{sin}\left(\frac{2\pi d}{C}\right),d\ge 0$

where $B$ and $C$ are positive constants.

According to the model, state what each of the constants $B$ and $C$ represents in this context.

According to the model,

• the model represents a normal (non-leap) year

• the maximum number of daylight hours is exactly $16$ hours and $38$ minutes

Find the total number of daylight hours in the first half of the year, from $d=0$ to $d=182.5$, giving your answer to the nearest $10$ hours.

Express

$115 \text{sin} \omega t+115\sqrt{3} \text{cos} \omega t$

in the form $R \text{sin}(\omega t+\alpha )$, where $R>0$ and $0<\alpha <\frac{\pi }{2}$.

Give the exact value of $R$ and the exact value of $\alpha$ in radians.

The alternating voltage, $V$ volts, in a domestic electrical circuit, $t$ seconds after an appliance is switched on, is modelled by the equation

$V=115 \text{sin} \omega t+115\sqrt{3} \text{cos} \omega t,t\ge 0$

where $\omega =2\pi f$ and $f$ is the frequency of the electricity in Hertz (Hz).

According to the model, the frequency of UK domestic electricity is exactly $50$ Hz.

(i) Find the exact voltage at the instant the appliance is switched on.

(ii) Find the first positive value of $t$ for which the voltage turns negative.

(i) Write down the period of one cycle of the alternating voltage in the UK.

According to the model, in the US, the period of one cycle of the alternating voltage is exactly $\frac{1}{60}$ seconds.

(ii) Write down the frequency of US domestic electricity.