The average daily hours of sunlight, S, during a 12-mon0wp f;xiqe.eo zb+w36th period in a particular town is given by to f;ip qxw w0be6.z+e3he function
S(t)=acos(bt−π)+d,where time, t, is measured in months, a and d are constants, and b is measured in degrees.
The graph of S versus t is shown in the diagram below.



1.Find the value of
(1)a;   
(2)d.   
2.Find the value of b.   $^{\circ}$
3.Write down the values of t when the average daily sunlight is 10 hours.
t =      

  The water depth, D, in metres, kt6c:e:c2gc ng 2m8)wed, o cvin a harbour on a particular day can be modelled by the equaevt 6wn) :2gkemcd o28gccc,:tion
D=2.5cos($30^{\circ}$$\times$t)+5, 0≤t≤24,
where t is the elapsed time, in hours, since midnight.
1.Draw the graph of D versus t on the grid below.

2.Find the lowest and highest depths of water in the harbour, and the times when they occur. $D_{min}$ =    t =       ; $D_{max}$ =    t =         
3.A large yacht has a draught of 3 metres. Decide whether the yacht will be able to enter the harbour at 5 pm.

  The water depth, D, in metr*b7u6j+e 5 2c2iw avxyat s,kaes, in a harbour on a particular can be modelled by the equatio6*5 ei2vja ck7xwu+b, sata2 yn
D=3sin($30^{\circ}$$\times$t)+5,0≤t≤24,
where t is the elapsed time, in hours, since midnight.
1.Draw the graph of D versus t on the grid below.

2.Find the lowest and highest depths of water in the harbour, and the times when they occur.  $D_{min}$ =    t =       ; $D_{max}$ =    t =   
3.A large boat has a draught of 2.5 metres. Decide whether the boat will be able to enter the harbour at 8 pm.

  The tip of a large wind turbine'tf-j*bm4wlf8o4bcen ;7ul,ni ),zv r s blade rotates at a constant speed on a windy day. Its height, H , in metres, above the groo, n)t ,u 8 bvmn47f4lbrceliz j-*fw;und is modelled by the function
H(t)=52sin($60^{\circ}\times$t)+80,
where t is the elapsed time, in seconds, since the turbine was accelerated to its max speed.
1.Write down the minimum height of the blade's tip above the ground. $H_{min}$ =   
2.Find the height of the blade's tip above the ground after 8 seconds.H(8 )≈   
3,Find the time it takes for the blade's tip to complete one full revolution.

  PM2.5 particles are tiny air poll9j.6sjin;e.jl n 6hxs: od;koutants with a diameter of 2.5 micrometres or less. In comparison, the average diameter of a human hair is about 70 micrometresmicrometr. ljon9di j .knsx e:os;;h6j6es. Air pollution from diesel & petrol engines form part of PM2.5 emissions and of particularparticular concern due to the health impacts.

During a particular working day, the rate of PM2.5 emissions, in grams per second, from all road vehicles in Beijing, China is modelled by the function
E(t)=6sin($\frac{π​}{6}$t−$\frac{5π​}{6}$)+9,0≤t≤24,
where t is the elapsed time, in hours, since midnight.
1.Find the maximum rate of PM2.5 emissions during the day in Beijing, China.$E_{max}$ =    g/sec
2.Find the times at which the maximum rate of PM2.5 emissions occur during the day. t =      

  Let $sinθ=-\frac{3}{5}$, for π<θ<$\frac{3π}{2}$​. Find the exact value of:
1.cosθ;c$\frac{a}{b}$ a =    b =    c =   
2.tanθ.c$\frac{a}{b}$ a =    b =    c =   

  Two identical water fswdm;e9 2gk(9 tu7t nwaves pass a sensor in an oceanography laboratory. As the wave 2d(s7tk;emwnu9fg t9 s pass the sensor, the depth, D, in metres, of water is recorded and modelled by the function
D(t)=3.21−0.45cos($24.2^{\circ}$$\times$t)
where t is the elapsed time, in seconds, since the first wave hit the sensor.
1.Find the minimum and maximum depths of the water as the two waves pass the sensor.$D_{min}$ =    ; $D_{max}$ =   
2.Find the first time after 1616 seconds at which the depth of water reaches 3.5m. t ≈   

  The Chicago Wheel, the world's first Ferris wheel, was designed and built q,5/w5j zuhda in 1893 by an American civil engineer, George W. G. Fer5h ,/wjqadz5 uris Jr.
The height, in metres, above the ground of a passenger on the Chicago Wheel after
t minutes can be modelled by the function
h(t)=38cos($\frac{π}{10}​(t−10))+42$.
1.Find the maximum height reached by a passenger on the Chicago Wheel.$h_{max}$ =   
2.Find the height above the ground of a passenger 12 minutes after the ride has started. h ≈   
3.Find the time, in minutes, it takes for the Chicago Wheel to complete one rotationrotation. t =    min
4.Given that passengers only complete one rotation on the Chicago Wheel, calculate how long they are more than 6565 metres above the ground. t ≈    min

  Let$cosθ=\frac{5}{6}$​, for $\frac{π}{2}$​<θ<π. Find the exact value of:
1.sinθ; a$\frac{\sqrt{b}}{c}$ a =    b =    c =   
2.tanθ.a$\frac{\sqrt{b}}{c}$ a =    b =    c =   

  The water depth, D, in metres, in a b.elpuv *vt*qt.0h rj 7oating marina on a particular day can be modelled by the fvt 0rjv.h7u*e*.qt plunction
D(t)​=4sin($30^{\circ}$$\times$t)+12.5,0≤t≤24,​where t is the elapsed time, in hours, since midnight.
1.Write down the depth of water at midnight.    m
2.The cycle of water depths repeats every T hours. Find the value of T.   
3.(1)Write down the minimum and maximum depths of water during the day.$D_{min}$ =    ; $D_{max}$ =   
(2)Find the times at which the minimum and maximum depths of water occur during the day.
4.Draw the graph of D versus t on the grid below.

5.A ship with 88 m draught is allowed to enter the marina if the depth of water is greater than 1010 m. Find the time interval before midday, in hours and minutes, during which the ship is not allowed to enter the marina.

  The water depth, D, in metres a*z5:lddhybonbtsh* )77 +if, at the entrance of a boating harbour can be modelle 7d il*yhsztn :ao*f7b+bd5 h)d by the function
D(t)​=4.5cos($30^{\circ}$$\times$t+$90^{\circ}$)+15,0≤t≤24,​where t is the elapsed time, in hours, since midnight.
1.Write down the depth of water at midnight.   
2.The cycle of water depths repeats every T hours. Find the value of T.   
3.(1)Write down the minimum and maximum depths of water during the day.$D_{min}$ =    ; $D_{max}$ =    (2)Find the times at which the minimum and maximum depths of water occur during the day.
4.Draw the graph of D versus t on the grid below.

5.A ship with 10 m draught is allowed to enter the harbour if the depth of waterwater is greater than 12 m. Find the time interval after midday, in hours and minutes, during which the ship is not allowed to enter the harbour.

  Let tanθ=$\frac{3}{4}$, for 0<θ<$\frac{π}{2}$​. Find the exact value of:
1.cosθ; $\frac{a}{b}$ a =    b =   
2.sinθ.$\frac{a}{b}$ a =    b =   

  The average petrol price, P, in cents per 0hbre mqxv 76d zc,8o,litre, in Australia 20192019 can be modelled by the functi6ormq,vh7e0d, cx8zbon
P(t)=asin($10^{\circ}C$$\times$t)+b,0≤t≤365,where t is the elapsed time, in days, since New Year's Eve 20182018.
Part of the graph of y=P(t) is shown below.


1.Write down the value of b.   
The average price of petrol on the 21st of January,2019 was 139139 cents per litre.
2.Show the value of a is equal to 12.   
3.Hence calculate:
(1)the average petrol price on the 1010th of February, 2019; P ≈    cents per litre
(2)the minimum and maximum average prices of petrol in Australia 2019.
The cycle of average petrol prices repeats every T days.$P_{min}$ ≈    cents per litre ; $P_{max}$ ≈    cents per litre
3.Determine the value of T.   

  The average gasoline price, P, in cents per galk1-+ a;1pcae v ztc(xslon, in USA 2019 can be modelled by the fu -a;1t+1k c(zxac pesvnction
P(t)=acos($15^{\circ}$t+$90^{\circ}$)+b,0≤t≤365,where t is the elapsed time, in days, since New Year's Eve 2018.
Part of the graph of y=P(t) is shown below.


1.Write down the value of b.  
The average price of gasoline on the 1414th of January, 2019 was 328 cents per gallon.
2.Show that the value of a is equal to 56.   
3.Hence calculate:
(1)the average gasoline price on the 1414th of February, 2019 ; P ≈    cents per gallon
(2)the minimum and maximum average prices of gasoline in USA 2019.
The cycle of average gasoline prices repeats every T days.$P_{min}$ ≈    cents per gallon; $P_{max}$ ≈    cents per gallon
4.Determine the value of T.   

  Let tanθ=$\frac{2}{3}$​, for $\frac{3π}{2}$​<θ<2π. Find the exact value of:
1.cosθ; a$\frac{b}{\sqrt{c}}$ a =    b =    c =   
2.sinθ.a$\frac{b}{\sqrt{c}}$ a =    b =    c =   

  ​

The frequency, f, in hertz (Hz), and the period, T, in seconds, of a wave are related by the equation
f=$\frac{1}{T}$​An electronic signal is sent out with amplitude 3 and frequency 200 Hz. The strength of this signal at time t seconds is given by
$S_1$​=3sin(bt)
1.Find the value of b, giving your answer in terms of π. b =    $π$
A second signal of equal strength is sent out with a time delay of 2 milliseconds. The strength of the second signal at time t seconds is given by
$S_2$​=3sin(b(t−c))
2.Write down the value of c.    seconds
The strength of the sum of the two signals at time t seconds is given by
$S_3$​=3sin(bt)+3sin(b(t−c))
3.Draw the graphs of $S_1$​, $S_2$​ and $S_3$​ versus t, for 0≤t≤0.01, on the same axes.

ABC is a triangle of nzp6ismn fjhl/;5, 0e area $32 cm^2$. The sides AB and BC have lengths 7 cm and 12 cm respectively. Find the two possible lengths of the side AC, giving your answers correct to 3 significant figures.

  The temperature, T, in degrees Celsius, in Istanbul during a particular winter dqx1(blm rg- 6d4e .,od;izbplay is modelled by the b 6eim,d4 q lb.;(ordgx -z1lpequation
T​=asin(b(t−c))+d,0≤t≤24,​where a,b,c,d∈R and t is the elapsed time, in hours, since midnight.
The graph of T versus t is shown below.




The lowest temperature was recorded at 3:30 am and the highest temperaturetemperature was recorded at 3:30 pm.
1.Write down the value of:
(1)a;   
(2)d.   
2.Find the value of b. ≈   
3.Find the smallest possible value of c, given c>0.
The temperature, T, in degrees Celsius, in Istanbul during a particular summer day is modelled by the equation
T​=5sin(0.262(t−8))+25,0≤t≤24,​where t is the elapsed time, in hours, since midnight. c =   
4.Find the time, in hours and minutes, when the temperature:
(1)reaches its maximum;
(2)first drops below $24^{\circ}C$.
5.The temperature is below $24^{\circ}C$ for ℎh hours and m minutes. Find
the value of h and the value of m. h =    m =   

  The tides at the Port ofe s8(hag1 .hb,2r xhjw Bristol, UK, were observed by a student.
On a particular day, the range between the lowest and highest tides is 10.6 m and the time difference between high tides is 12.2 hours. The first highest tide occurs at 8:12 am and is 12.5 m high.
The height, H, in metres, of the tides can be modelled by the equation
H​=asin(b(t−c))+d,0≤t≤24,​where t is the elapsed time, in hours, since midnight.
1.Write down the value of:
(1)a;   
(2)d.   
2.Find the value of b. ≈   
3.Find the smallest possible value of c, given c>0. ≈   
4.Hence draw the graph of H versus t, for 0≤t≤24.

Let tanθ=r, for $\frac{π}{2}​$<θ<$π$, where r is a negative constant.
Express in terms of r:
1.cosθ;
2.sinθ.

Let tanθ=k, for $\frac{3π}{2}​$<θ<$2π$, where k is a negative constant.
Express in terms of r:
1.cosθ;
2.sinθ.

  

  The temperature in New York rangea q.x/.* 7 tlhiw8(oma+i7n0pn 2mmosnmjz8r s from $-4^{\circ}C$ to $22^{\circ}C$ throughout the course of a year. The temperature, T, can be modelled by the function
T(t)=−acos(bt)+d,
Where t is time measured in months from the beginning of the year, and a, b and d∈Z+.
1.Show that
(1)a=13;   
(2)b=30;   
(3)d=9.   
2.Sketch the function 0≤t≤12, clearly labelling the coordinates of the maximum and minimum points.
3.Find the temperature when t=4.
4.(1)Determine the number of months in a year that the temperature is above $^{\circ}C$.
(2)Find the probability that on a randomly chosen day of the year, the temperature is above $9^{\circ}C$.
5.If the temperature in New York ranged from $-7^{\circ}C$to $25^{\circ}C$ instead of $-4^{\circ}C$ to $22^{\circ}C$, describe how this would affect the probability found in (d) part (ii).

  The amount of water, A, in thousands of litres, available in a ,ynfy2q)c ck:water tank located on a farm fluctuates in a yearly cycle and can be modell2 ,qc )fc:knyyed by the function
A(t)=asin(kt)+b,where t is the elapsed time, in weeks, since the start of the year.
The amount of water available in the tank on week 66 is 24 thousand litres and on week 31 is 9.2 thousand litres.
1.Find the value of k, in degrees, assuming there are 52 whole weeks in a year. ≈    $^{\circ}$
2.Set up a pair of equations to find the value of a and the value of b. Give your answers correct to the nearest integer. a ≈    b ≈   
3.Hence find the amount of water available in the tank in week 42.    thousands liters
4.Draw the graph of y=A(t) on the grid below, for one full year, indicating clearly the minimum and maximum points.

  The mean depth, D, in metres)z4cbtly yv 83, of a mountain lake fluctuates in a yearly cycle and can b3b8zy)l4t v yce modelled by the function
D(t)=acos(kt)+b,where t is the elapsed time, in months, since the beginning of an autumn season.
The mean depth of the lake on month 11 is 33.2 m and on month 5 is 22.8 m.
1.Find the value of k, in degrees.    $^{\circ}$
2.Set up a pair of equations and find the value of a and the value of b. Give your answers correct to the nearest integer. a ≈    b ≈   
3.Hence find the mean depth of the lake on month 8.    m
4.Draw the graph of y=D(t) on the grid below, for one full year, indicating clearly the minimum and maximum points.

  Alexandros monitors the temperature d89:4cc iytncpi+:(p,q f4u a7gwrlp muring a particular day in Athens.
The table below shows some of his findings.


Based on his observations, Alexandros concludes that the temperature, T, in $^{\circ}C$, during the day can be modelled by the equation
T​=asin(b(t−c))+d,0≤t≤24,​where t is the elapsed time, in hours, since midnight.
1.Write down the value of:
(1)a;   
(2)d.   
2.Find the value of b. ≈   
3.Find the smallest possible value of c, given c>0.   
4.Find the time, in hours and minutes, when the temperature first reaches $35^{\circ}C$ during the day.

Two voltage sources are cgsg 1/ z*il+qhlz7 )m yn-zl.jonnected to a circuit. At time t milliseconds (ms), the voltage from the first sou +-*jzhgs.qg 7/ll1m)zilyn zrce is $V_1$​(t)=12cos(20t) and the voltage from the second source is $V_2$​(t)=18cos(20t+5), where both $V_1$​(t) and $V_2$​(t) are measured in volts.
1.Write, in the form V(t)=Acos(ωt+φ), an expression for the total voltage in the circuit at time t ms.
2.Hence write down the highest voltage in the circuit.

  Annika, Bob, Chloe and Dani go to a fair and take a ride on a F)*( z5hseacst erris wheel. The wste5)sh*a(c z heel has
8 evenly spaced passenger cars as shown in the following diagram. The wheel completes one rotation in the anticlockwise direction every 80 seconds



Annika starts her ride at position A. Let $h_A$​ be Annika's height in meters after t seconds. $h_A$​​ can be modelled by the function
$h_A$​(t)=asin(bt)+d,where a,b,d∈R.​
The diagram below shows the graph of y=$h_A$​(t) for one revolution.




1.Show that
(1)a=14.
(2)b=$\frac{π}{40}$​.
(3)d=18.
At the time Annika starts her ride, Bob is at position B. Bob's position can be modeled by the function $h_B$​(t)=14cos($\frac{π}{40}$​t)+18.
2.Determine
(1)the first time at which Annika and Bob are at the same height.
(2)the height at which this occurs.
At the time Annika starts her ride, Chloe is at position C. Chloe's position at time t can be written as $h_C$​(t)=m(hA​(t))+n.
3.Find the value of m and the value of n.
When Chloe reaches an altitude of 3030 meters she has a view of the whole town.   
4.(1)Find the time when Chloe has a view of the whole town for the first time.
(2)Find the angle rotated through by Chloe's car from its start position to the point when she has a view of the whole town for the first time. Give your answer in radians, correct to one decimal place.
Dani is at position D when Annika starts her ride. ≈    radians
5.Find the value of c such that the function $h_D$​(t)=14sin($\frac{π}{40}$​(t−c))+18 describes Dani's height at time t.
The function D(t) represents the difference in height between Annika and Dani's cars. c =   
6.Write D(t) as the difference of two sine functions.
D(t) can be written in the form Im($z_1​−z_2$​), where $z_1$​ and $z_2$​ are complex functions of t.
7.(1)Write $z_1$​ and $z_2$​ in exponential form.
(2)Hence or otherwise find an equation for D(t) in the form D(t)=psin(qt+r)+s, where p,q,r,s∈R.
(3)Find the maximum difference in height between Annika and Dani's cars.

In an unbalanced three-phase electrical circuit, the current at time t ms is gmfktesob9: 9gh982 nziven byf ozs:b9ge29n m k89th
I(t)= 2sin(5t)+5sin(5t-$\frac{3π}{4})+10sin(5t-$\frac{5π}{4}$)
where I(t) is measured in milliamperes (mA).
1.Write I(t) in the form Acos(ωt+φ).
2.Hence find the highest current flowing through the circuit, and the time it first occurs.

  The revenues of a four sekzs h ej+p;n b4+l/fjhk/3p*xasons hotel can be modelled by the function
R(t)=58.2sin(0.0172t−1.25)+204,
where t is the number of days after midnight on 31 December.
In a similar way, the operating costs of the hotel can be modelled by the function
C(t)=31.4sin(0.0172t+1.14)+85.0.
Both R(t) and C(t) are measured in thousand dollars.
1.Show that the profits of the hotel can be modelled by the function P(t)=83.9sin(0.0172t−1.51)+119.
2.According to the model, find:
(1)the highest profit the hotel will make; ≈    thounds
(2)the date on which the highest profit will occur.

Ali is swimming in a public pool with some of his friends. At time t seconds,a 6 mx,+mrcxv6 he encount6x c,mm+6ra xversencounters some waves with height $h_1$​(t)=0.15sin(3t) from big Bobby jumping into the pool, and waves of height $h_2$​(t)=0.08sin(3t+1.25) from small Suzie jumping into the pool. Both $h_1$​(t) and $h_2$​(t) are measured in metres.
1.Write, in the formh(t)=Asin(ωt+φ), an expression for the total height of the waves Ali encounters at time t seconds.
2.Find the times in the first 5 seconds when Ali isn't affected by any waves.
3.Find the first time when the waves reaching Ali has maximum height.