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Trigonometry, 2ADV T3 2025 HSC 15

A sound wave can be modelled using a function  \(P(t)=k\, \sin a t\), where \(P\) is air pressure in Pascals, \(t\) is time in milliseconds (ms) and \(k\) and \(a\) are constants.

  1. Write the equation for a sound wave \(P_1(t)\) that has an amplitude of 2 Pascals and a period of 5 ms.   (2 marks)

  2. The graph of \(P_1(t)\) from part (a) is shown.
  3. On the diagram, sketch the graph of  \(P_2(t)=4 \sin \left(\dfrac{\pi}{10} t\right)\)  for  \(0 \leq t \leq 10\).   (2 marks)
     

  1. Hence, find the values of \(t\), where  \(0<t<10\),  for which functions \(P_1(t)\) and \(P_2(t)\) are BOTH decreasing.   (2 marks)

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a.  \(P_1(t)=2\, \sin \left(\dfrac{2 \pi t}{5}\right)\)

b.   
       

c.  \(\text{Both decreasing for} \ \ 6.25<t<8.75\)

Show Worked Solution

a.    \(\text{Amplitude}=2 \Rightarrow k=2\)

\(\text{Period}=5\)

\(\dfrac{2 \pi}{a}\) \(=5\)
\(5a\) \(=2 \pi\)
\(a\) \(=\dfrac{2 \pi}{5}\)

 
\(\therefore P_1(t)=2\, \sin \left(\dfrac{2 \pi t}{5}\right)\)
 

b.   
     

\(P_2(t)=4\, \sin \left(\dfrac{\pi}{10} t\right)\)

\(\text{Amplitude}=4\)

\(\text{Period}=\dfrac{2 \pi}{\frac{\pi}{10}}=20\)
 

c.    \(\text {By inspection of graph:}\)

\(P_2(t) \ \text {is decreasing for} \ \ 5<t \leq10\)

\(P_1(t) \text { is decreasing for} \ \ 1.25<t<3.75 \ \ \text{and}\ \ 6.25<t<8.75\)

\(\therefore \ \text{Both decreasing for} \ \ 6.25<t<8.75\)

Trigonometry, 2ADV T3 EQ-Bank 4

The graph below has the equation  \(y=a \sin (b x)+c\)  for  \(0 \leqslant x \leqslant 50\).
 

Determine the values of  \(a, b\)  and  \(c\).  (3 marks)

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\(a=16, b=9, c=24\)

Show Worked Solution

\(\text {Amplitude}\ =\dfrac{40-8}{2}=16\ \ \Rightarrow a=16\)

\(\text {Centre of motion}\ =24\ \ \Rightarrow c=24\)

\(\text {Period }=\dfrac{360}{n}=40\ \ \Rightarrow n=b=9\)

Trigonometry, 2ADV T3 2022 HSC 14

The graph of  `y=k sin (ax)`

What are the values of `k` and `a`?  (2 marks)

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`k=4, \ \ a=1/3`

Show Worked Solution

`text{Amplitude = 4}`

`=> k=4`

`text{Period} = 6pi`

`(2pi)/a` `=6pi`  
 `6pia` `=2pi`  
 `=>a` `=1/3`  

Trigonometry, 2ADV T3 2021 HSC 20

For what values of `x`, in the interval  `0 <= x <= pi/4`,  does the line  `y = 1`  intersect the graph of  `y = 2sin4x`?  (2 marks)

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`pi/24, (5pi)/24`

Show Worked Solution

`text(Find)\ x\ text(such that:)`

`2sin4x` `= 1`
`sin4x` `= 1/2`
`4x` `= sin^(-1)\ 1/2`
`4x` `= pi/6, (5pi)/6, (13pi)/6, (17pi)/6, …`
`:. x` `= pi/24, (5pi)/24\ \ \ (0 <= x <= pi/4)`

Trigonometry, 2ADV T3 EQ-Bank 5

The function  `f(x) = sin x`  is transformed into the function  `g(x) = (sin(4x))/3`.

Describe in words how the amplitude and period have changed in this transformation.  (2 marks)

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`text(See Worked Solutions)`

Show Worked Solution

`g(x) = 1/3 sin (4x)`

`=>\ text(The new amplitude is one third of the original amplitude.)`

`text(Period)\ = (2pi)/n \ => \ \ n=1/4`

`=>\ text(The new period is one quarter of the original period.)`

Trigonometry, 2ADV T3 2019 HSC 7 MC

The diagram shows part of the graph of  `y = a sin(bx) + 4`.
 

What are the values of `a` and `b`?

  1. `a = 3 qquad qquad b = 1/2`
  2. `a = 3 qquad qquad b = 2`
  3. `a = 1.5 qquad \ b = 1/2`
  4. `a = 1.5 qquad \ b = 2`
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`D`

Show Worked Solution

`a = 1/2 (5.5 – 2.5) = 1.5`

`text(S)text(ince graph passes through)\ \ (pi/4, 5.5):`

`5.5 = 1.5 sin(b xx pi/4) + 4`

`sin (b xx pi/4)` `= 1`
`b xx pi/4` `= pi/2`
`:. b` `= 2`

 
`=>  D`

Trigonometry, 2ADV T3 SM-Bank 8

`f(x) = 2 sin (2x)`  is defined in the domain  `{x: \ pi/8 <= x < pi/3)`

What is the range of the function  `f(x)`?  (2 marks)

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`text(Range) = [sqrt 2, 2]`

Show Worked Solution

♦ Mean mark 45%.

`sin(2x)_text(max)\ \ text(occurs when)\ \ x=pi/4\ \ text{(within domain)}`

`=> f(x)_text(max)  = 2 sin(pi/2) = 2`

`text(Checking endpoints:)`

`text(When)\ \ x=pi/8\ \ =>\ \ y=2 sin(pi/4) = sqrt2`

`text(When)\ \ x=pi/3\ \ =>\ \ y=2 sin((2pi)/3) = sqrt3`
 


 

`:.\ text(Range) = [sqrt 2, 2],`

Trigonometry, 2ADV T3 SM-Bank 3 MC

Let  `f (x) = 5sin(2x) - 1`.

The period and range of this function are respectively

  1. `π\ text(and)\ [−1, 4]`
  2. `2π\ text(and)\ [−1, 5]`
  3. `π\ text(and)\ [−6, 4]`
  4. `2π\ text(and)\ [−6, 4]`
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`C`

Show Worked Solution

`text(Period) = (2pi)/2 = pi`

`text(Range)` `= [−1 – 5, −1 + 5]`
  `= [−6 ,4]`

 
`=> C`

Trigonometry, 2ADV T3 SM-Bank 1 MC

`f(x) = 2sin(3x) - 3`

The period and range of this function are respectively

  1. `text(period) = (2 pi)/3 and text(range) = text{[−5, −1]}`
  2. `text(period) = (2 pi)/3 and text(range) = text{[−2, 2]}`
  3. `text(period) = pi/3 and text(range) = text{[−1, 5]}`
  4. `text(period) = 3 pi and text(range) = text{[−1, 5]}`
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`A`

Show Worked Solution

`text(Range:)\ [−3 -2, −3 + 2]`

`= [−5,−1]`

`text(Period) = (2pi)/n = (2pi)/3`

`=>   A`

Trigonometry, 2ADV T3 2010 HSC 8c

The graph shown is  `y = A sin bx`.

 

  1. Write down the value of  `A`.    (1 mark)

  2. Find the value of  `b`.    (1 mark)

  3. On the same set of axes, draw the graph  `y = 3 sin x + 1`  for  `0 <= x <= pi`.   (2 marks)

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  1. `A = 4`
  2. `b = 2`
  3. `text(See Worked Solutions for sketch)`
Show Worked Solution

i.   `A = 4`

ii.  `text(S)text(ince the graph passes through)\ \ (pi/4, 4)`

`text(Substituting into)\ \ y = 4 sin bx`

`4 sin (b xx pi/4)` `=4`
`sin (b xx pi/4)` `= 1`
`b xx pi/4` `= pi/2`
`:. b` `= 2`

  

 MARKER’S COMMENT: Graphs are consistently drawn too small by many students. Aim to make your diagrams 1/3 to 1/2 of a page. 
iii.

Trigonometry, 2ADV T3 2013 HSC 6 MC

Which diagram shows the graph  `y=sin(2x + pi/3)`?
 

2013 6 mc1

2013 6 mc2

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`D`

Show Worked Solution
♦♦ Mean mark 34%

`text(At)\ x = 0 text(,)\ \ y = sin (pi/3)  = sqrt3/2`

`=>\ text(It cannot be A or C)`
 

`text(Find)\ x\ text(when)\ y = 0,`

`sin (2x + pi/3)` `= 0`
`:.\ 2x + pi/3` `= 0\ \ \ \ \ text{(sin 0 = 0)}`
`2x` `=-pi/3`
`x` `= -pi/6`

`=>  D`