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Trigonometry, 2ADV T3 2025 MET2 1 MC

A function that has a range of \([6,12]\) is

  1. \(f(x)=6+3 \cos (9 x)\)
  2. \(f(x)=6+6 \cos (3 x)\)
  3. \(f(x)=9-3 \cos (6 x)\)
  4. \(f(x)=9-6 \cos (3 x)\)
Show Answers Only

\(C\)

Show Worked Solution

\(\text{By trial and error,}\)

\(\text{Consider option C:}\ \  f(x)=9-3 \cos (6 x)\)

\(\text{Since}\ \ -1 \leqslant \cos (6 x) \leqslant 1\)

\(\text{Range is} \ \ [-3+9,3+9]=[6,12]\)

\(\Rightarrow C\)

Trigonometry, 2ADV T3 2025 MET1 3

Let  \(f(x)=2 \cos (2 x)+1\)  over the domain \(x \in\left[0, 2 \pi \right]\).

  1. State the range of \(f(x)\).   (1 mark)

  2. Solve  \(f(x)=0\)  for \(x\).   (3 marks)

  3. Sketch the graph of  \(y=f(x)\)  for  \(x \in\left[\dfrac{\pi}{2}, \dfrac{3 \pi}{2}\right]\) on the axes below.
  4. Label the endpoints with their coordinates.   (2 marks)

Show Answers Only

a.    \(\text{Range of } f(x):-1 \leqslant y \leqslant 3\)

b.    \(x=\dfrac{\pi}{3}, \dfrac{2 \pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}\)

c.   

   

Show Worked Solution

a.    \(\text{Amplitude}=2 \ \ \text{about} \ \ y=1.\)

\(\text{Range of } f(x):\ -1 \leqslant y \leqslant 3\)
 

b.     \(2 \cos (2 x)+1\) \(=0\)
  \(\cos (2 x)\) \(=-\dfrac{1}{2}\)

 
\(\text{Base angle}=\dfrac{\pi}{3}\)

\(2x=\pi-\dfrac{\pi}{3}, \pi+\dfrac{\pi}{3}, \cdots\)

\(2x=\dfrac{2 \pi}{3}, \dfrac{4 \pi}{3}, \dfrac{8 \pi}{3}, \dfrac{10 \pi}{3}\)

  \(x=\dfrac{\pi}{3}, \dfrac{2 \pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}\)
  

c.   

   

♦ Mean mark (c) 49%.

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)

Show Answers Only

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 28

A function is defined as  \(f(x)=-2\cos\Big( \dfrac{\pi x}{3} \Big). \)

  1. Determine the amplitude and period of the function.   (2 marks)
  2. Sketch the function over the domain  \( {0 \leq x \leq 2\pi}\). On your sketch, label any \(x\)-axis intersections.   (3 marks)   
Show Answers Only

a.    \(\text{Amplitude}\ = 2\)

\(\text{Period}\ (T) = 6 \)

b.    
         

Show Worked Solution

a.    \(\text{Amplitude}\ = 2\)

\(\text{Period}\ (T) = \dfrac{2\pi}{n} = \dfrac{2\pi}{\frac{\pi}{3}} = 6 \)
 

b.   
             

Trigonometry, 2ADV T3 EQ-Bank 18

  1. Sketch the function \(y=\cos \left(\dfrac{x}{2}\right)\)  from  \(0 \leqslant x \leqslant 2 \pi\)   (1 mark)

     

  2. Find the value of \(x\) when  \(y=0.5\), for  \(0 \leqslant x \leqslant 2 \pi\)   (2 marks)

Show Answers Only

a.    
           

b.   \(x=\dfrac{2\pi}{3}\)

Show Worked Solution

a.    
           

b. \(\quad y\) \(=\cos \Big(\dfrac{x}{2}\Big)\)
  \(0.5\) \(=\cos \Big(\dfrac{x}{2}\Big) \)
  \(\dfrac{x}{2}\) \(=\cos^{-1} (0.5) \)
  \(\dfrac{x}{2}\) \(=\dfrac{\pi}{3}, \ \dfrac{5\pi}{3}\)
  \(x\)  \(=\dfrac{2\pi}{3}\ \ (0 \leqslant x \leqslant 2 \pi) \)

Trigonometry, 2ADV T3 EQ-Bank 22

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)

Show Answers Only

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

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 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)

Show Answers Only

`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 2010 MET1 3

Shown below is part of the graph of a period of the function of the form  `y = tan(ax + b)`.
 


 

Find the value of `a` and the value of `b`, where  `a > 0`  and  `0 < b < 1`.   (3 marks)

Show Answers Only

`a = (7pi)/24,\ \ b = pi/24`

Show Worked Solution

`y = tan(ax + b)`

`text(Substitute)\ \ (1, sqrt3), (−1, −1)\ \ text(into equation:)`

`tan(a + b)` `= sqrt3`
`tan(b-a)` `=-1`
`a + b` `= pi/3 \ …\ (1)`
`b-a` `=-pi/4 \ …\ (2)`

 
`text{Add (1) + (2):}`

`2b` `= pi/3-pi/4`
`b` `= pi/24`

 
`text{Substitute into (1):}`

`a + pi/24` `= pi/3`
`a` `= (7pi)/24`

Trigonometry, 2ADV T3 EQ-Bank 32

By drawing graphs on the number plane, show how many solutions exist for the equation  `cosx = |(x-pi)/4|`  in the domain  `(-∞, ∞)`   (3 marks)

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`text(4 solutions)`

Show Worked Solution

`text(Sketch:)`

`y` `= cos x`
`y` `= |(x-pi)/4|`

 
`text(Translate)\ pi\ text(units to the right: )`

`y=|x| \ => \ y=|x-pi|`

`text(Multiply by)\ 1/4 :`

`y=|x-pi| \ => \ y= 1/4 |x-pi| = |(x-pi)/4|`

`:.\ text(There are 4 solutions.)`

Trigonometry, 2ADV T3 EQ-Bank 12

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)

Show Answers Only

`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 EQ-Bank 4 MC

The diagram below shows one cycle of a circular function.
 

The amplitude and period of this function are respectively

  1. `3\ \ text(and)\ \ 2 `
  2. `3\ \ text(and)\ \ (pi)/(2)`
  3. `4\ \ text(and)\ \ (pi)/(4)`
  4. `3\ \ text(and)\ \ 4`
Show Answers Only

`D`

Show Worked Solution

`text(Graph centres around)\ \ y = 1`

`text(Amplitude) \ = 3`

`text(Period) = 4`

`=> D`

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`
Show Answers Only

`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 EQ-Bank 29

For the function  `f(x) = 5 cos (2 (x + pi/3)),\ \ \ -pi<=x<=pi`

  1. Write down the amplitude and period of the function.   (2 marks)

  2. Sketch the graph of the function  `f(x)`  on the set of axes below. Label axes intercepts with their coordinates.

     

    Label endpoints of the graph with their coordinates.   (3 marks)

VCAA 2006 meth 4b

Show Answers Only

a.    `text(Amplitude) = 5;\ \ \ text(Period) = pi`

b.  

 

Show Worked Solution

a.    `text(Amplitude) = 5`

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

 

b.    `text(Shift)\ \ y = 5 cos (2x)\ \ text(left)\ \ pi/3\ \ text(units).`

`text(Period) = pi`

`text(Endpoints are)\ \ (-pi, -5/2) and (pi,-5/2)`

Trigonometry, 2ADV T3 EQ-Bank 23

State the range and period of the function

`h(x) = 4 + 3 cos ((pi x)/2).`   (2 marks)

Show Answers Only

`text(Range:)\ \ 1<=y<= 7`

`text(Period) = 4`

Show Worked Solution
  `-1` `<= cos ((pi x)/2)<=1`
  `-3` `<=3cos ((pi x)/2)<=3`
  `1` `<= 4+ 3cos ((pi x)/2)<=7`

 
`:.\ text(Range:)\ \ 1<=y<= 7`

 
`text(Period) = (2pi)/n = (2 pi)/(pi/2) = 4`

Trigonometry, 2ADV T3 EQ-Bank 19

Let   `f(x) = 2cos(x) + 1`  for  `0<=x<=2pi`.

  1. Solve the equation  `2cos(x) + 1 = 0`  for  `0 <= x <= 2pi`.   (2 marks)

  2. Sketch the graph of the function  `f(x)`  on the axes below. Label the endpoints and local minimum point with their coordinates.   (3 marks)

 

Show Answers Only

a.    `(2pi)/3, (4pi)/3`

b.
     

Show Worked Solution
a.    `2cos(x) + 1` `= 0`
  `cos(x)` `=-1/2`

`=> cos\ pi/3 = 1/2\ text(and cos is negative)`

`text(in 2nd/3rd quadrant)`

`:.x` `= pi-pi/3, pi + pi/3`
  `= (2pi)/3, (4pi)/3`

 

b.    

Trigonometry, 2ADV T3 EQ-Bank 33

`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)

Show Answers Only

`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 EQ-Bank 8 MC

met2-2011-vcaa-15-mc 
 

The graph shown could have equation

  1. `y = 2cos(x + pi/6) + 1`
  2. `y = 2cos4(x-pi/6) + 1`
  3. `y = 4sin2(x-pi/12)-1`
  4. `y = 3cos(2x + pi/6)-1`
Show Answers Only

`B`

Show Worked Solution

`text{Amplitude = 2   (range from – 1 to 3)}`

`text{Graph centre line (median):}\ \ y= 1`

`:.\ text(Eliminate)\ \ C\ \ text(and)\ \ D.`
 

`text(Period) = (2pi)/3-pi/6 = pi/2\ \ text{(from graph)}`

`text(Consider option)\ B,`

`text(Period)= (2pi)/n= (2pi)/4 = pi/2`

`=> B`

Trigonometry, 2ADV T3 EQ-Bank 2 MC

Let   `f(x) = 1-2 cos ({pi x}/2).`

The period and range of this function are respectively

  1. `4 and [-2, 2]`
  2. `4 and [-1, 3]`
  3. `1 and [-1, 3]`
  4. `4 pi and [-2, 2]`
Show Answers Only

`B`

Show Worked Solution

`text(Period)= (2 pi)/n = (2pi)/(pi/2)=4`

`text(Amplitude = 2)`

`text{Graph centre line (median):}\ \ y=1.`

`:.\ text(Range)` `= [1-2, quad 1 + 2]`
  `= [-1, 3]`

`=>   B`

Trigonometry, 2ADV T3 EQ-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) = [-5,-1]`
  2. `text(period) = (2 pi)/3 and text(range) = [-2, 2]`
  3. `text(period) = pi/3 and text(range) = [-1, 5]`
  4. `text(period) = 3 pi and text(range) = [-1, 5]`
Show Answers Only

`A`

Show Worked Solution

`text(Range:)\ [-3 -2,\  -3 + 2]`

`= [-5,-1]`

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

`=>   A`

Trigonometry, 2ADV T3 2006 HSC 7b

A function  `f(x)`  is defined by  `f(x) = 1 + 2 cos x`.

  1. Show that the graph of  `y = f(x)`  cuts the `x`-axis at  `x = (2 pi)/3`.   (1 mark)

  2. Sketch the graph of  `y = f(x)`  for  `-pi <= x <= pi`  showing where the graph cuts each of the axes.   (3 marks)

  3. Find the area under the curve  `y = f(x)` between  `x = -pi/2`  and  `x = (2 pi)/3`.   (3 marks)

Show Answers Only

a.    `text(Proof)\ \ text{(See Worked Solutions)}`

b.    
    

c.    `((7 pi)/6 + sqrt 3 + 1)\ text(u²)`

Show Worked Solution

a.    `f(x) = 1 + 2 cos x`

`f(x)\ text(cuts the)\ x text(-axis when)\ f(x) = 0`

`1 + 2 cos x` `= 0`
`2 cos x` `=-1`
 `cos x` `=-1/2`

`:.  x = (2 pi)/3\ …\ text(as required)`
  

b.     2UA HSC 2006 7b

 

c.    `text(Area)` `= int_(-pi/2)^((2 pi)/3) 1 + 2 cos x\ \ dx`
  `= [x + 2 sin x]_(-pi/2)^((2 pi)/3)`
  `= [((2 pi)/3 + 2 sin­ (2 pi)/3)-((-pi)/2 + 2 sin­ (-pi)/2)]`
  `= ((2 pi)/3 + 2 xx sqrt 3/2)-((-pi)/2 +2(- 1))`
  `= (2 pi)/3 + sqrt(3) + pi/2 + 2`
  `= ((7 pi)/6 + sqrt(3) + 2)\ text(u²)`

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)

Show Answers Only

a.    `A = 4`

b.    `b = 2`

c.    `text(See Worked Solutions for sketch)`

Show Worked Solution

a.    `A = 4`

b.    `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. 
c.   

Trigonometry, 2ADV T3 2013 HSC 6 MC

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

2013 6 mc1

2013 6 mc2

Show Answers Only

`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`