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Trigonometry, EXT1 T1 2025 HSC 11d

Sketch the graph of  \(y=\dfrac{1}{3} \cos ^{-1}(2 x)\).   (2 marks)

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\(y=\dfrac{1}{3} \cos ^{-1}(2 x)\)

\(\text{Domain:} \ -1 \leqslant 2 x \leqslant 1 \ \Rightarrow \ -\dfrac{1}{2} \leqslant x \leqslant \dfrac{1}{2}\)

\(\text{Range:} \ \cos (2 x) \in[0, \pi] \  \Rightarrow \  \dfrac{1}{3} \cos (2x) \in\left[0, \dfrac{\pi}{3}\right]\)
 

Trigonometry, EXT1 T1 2024 HSC 5 MC

Consider the function  \(g(x) = 2 \sin^{-1}(3x)\).

Which transformations have been applied to \(f(x) = \sin^{-1}(x)\)  to obtain \(g(x)\)?

  1. Vertical dilation by a factor of \(\dfrac{1}{2}\) and a horizontal dilation by a factor of \(\dfrac{1}{3}\)
  2. Vertical dilation by a factor of \(\dfrac{1}{2}\) and a horizontal dilation by a factor of 3
  3. Vertical dilation by a factor of 2 and a horizontal dilation by a factor of \(\dfrac{1}{3}\)
  4. Vertical dilation by a factor of 2 and a horizontal dilation by a factor of 3
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\(C\)

Show Worked Solution

\(\text{A vertical dilation of factor 2:}\)

\(f(x) = \sin^{-1}(x)\ \ \rightarrow \ \ f_1(x) = 2\sin^{-1}(x)\)

\(\text{A horizontal dilation of factor}\ \dfrac{1}{3}:\)

\(f_1(x) = 2\sin^{-1}(x)\ \ \rightarrow \ \ f_2(x) = 2\sin^{-1}(3x)\)

\(\Rightarrow C\)

Trigonometry, EXT1 T1 2024 HSC 4 MC

What are the domain and range of the function  \(y = 2 \cos^{-1}(2x) + 2 \sin^{-1}(2x)\)?

  1. Domain: \([-0.5, 0.5]\) and Range: \(\{\pi\}\)
  2. Domain: \([-0.5, 0.5]\) and Range: \([-\pi, 3 \pi ]\)
  3. Domain: \([-2, 2]\) and Range: \(\{\pi\}\)
  4. Domain: \([-2, 2]\) and Range: \([-\pi, 3\pi]\)
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\(A\)

Show Worked Solution

\(\text{Domain:}\ \ -1 \leqslant 2x \leqslant 1 \ \ \Rightarrow\ \ -\dfrac{1}{2} \leqslant x \leqslant \dfrac{1}{2} \)

\(\text{Range:}\ \ 2\Big(\cos^{-1}(2x)+ \sin^{-1}(2x)\Big) = 2 \times \dfrac{\pi}{2} = \pi\)

\(\Rightarrow A\)

Trigonometry, EXT1 T1 EQ-Bank 17

  1. State the domain and range for  \(f(x)=4 \sin ^{-1}(2 x-3)\).   (2 marks)

  2. Sketch the function  \(f(x)=4 \sin ^{-1}(2 x-3)\).  (1 mark)

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a.    \(\text{Domain:}\ \ -1 \leqslant\) \(2 x-3 \leqslant 1 \)
  \( 2 \leqslant\) \(2 x \leqslant 4\)
  \( 1 \leqslant\)  \(x \leqslant 2\)

 

        \(\text{Range:}\ \ -\dfrac{\pi}{2} \leqslant\) \(y \leqslant \dfrac{\pi}{2}\)
  \( -2 \pi \leqslant \) \(4 y \leqslant 2 \pi\)

 
b.

Show Worked Solution

a.    \(\text{Domain:}\ \ -1 \leqslant\) \(2 x-3 \leqslant 1 \)
  \( 2 \leqslant\) \(2 x \leqslant 4\)
  \( 1 \leqslant\)  \(x \leqslant 2\)

 

        \(\text{Range:}\ \ -\dfrac{\pi}{2} \leqslant\) \(y \leqslant \dfrac{\pi}{2}\)
  \( -2 \pi \leqslant \) \(4 y \leqslant 2 \pi\)

 
b.

Trigonometry, EXT1 T1 EQ-Bank 7 MC

Which equation describes the graph shown?
 

  1. \(y=2 \cos ^{-1}(x+1)\)
  2. \(y= \sin ^{-1}(x-1)\)
  3. \(y=2 \cos ^{-1}(x-1)\)
  4. \(y=\dfrac{1}{2} \sin ^{-1}(x+1)\)
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\(C\)

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\(\text{Graph shape (general)}\ \ → \cos^{-1}(x) \)

\(\text{Graph translated 1 unit to the right}\ → \cos^{-1}(x-1) \)

\(\text{Range:}\ 0 \leq \cos^{-1}(x-1) \leq \pi\ \ →\ 0 \leq 2\cos^{-1}(x-1) \leq 2\pi \)

\(\Rightarrow C\)

Trigonometry, EXT1 T1 EQ-Bank 22

Sketch the graph  \(y=2 \cos ^{-1}(x+1)\).   (3 marks)

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\(\text{Domain:}\)

\(-1 \leqslant x+1 \leqslant 1 \ \Rightarrow \ -2 \leqslant x \leqslant 0\)

\(\text{Range:}\)

\(0 \leqslant \cos ^{-1}(x) \leqslant \pi\ \ \Rightarrow \ \ 0\leqslant 2 \cos ^{-1}(x) \leqslant 2 \pi\)
 

Trigonometry, EXT1 T1 EQ-Bank 11

State the domain and range of the function

\(y=\arccos \, 3x\)   (2 marks)

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\(\text{Domain:}\ \ \Big[-\dfrac{1}{3}, \dfrac{1}{3}\Big] \)

\(\text{Range:}\ \ [0, \pi] \)

Show Worked Solution

\(\text{Consider domain:}\ \ -1 \leqslant 3x \leqslant 1 \ \Rightarrow\ -\dfrac{1}{3} \leqslant x \leqslant \dfrac{1}{3}\)

\(\text{Domain:}\ \ \Big[-\dfrac{1}{3}, \dfrac{1}{3}\Big] \)

\(\text{Range:}\ \ [0, \pi] \)

Trigonometry, EXT1 T1 2023 HSC 7 MC

Which statement is always true for real numbers \(a\) and \(b\) where \(-1 \leqslant a<b \leqslant 1\)?

  1. \(\sec a<\sec b\)
  2. \(\sin ^{-1} a<\sin ^{-1} b\)
  3. \(\arccos a<\arccos b\)
  4. \(\cos ^{-1} a+\sin ^{-1} a<\cos ^{-1} b+\sin ^{-1} b\)
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\(B\)

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\(\text{Consider the graph of}\ \ y=\sin^{-1}x \)
 

\( y=\sin^{-1}x\ \ \text{is an increasing function in range}\ \ -1 \leqslant x \leqslant 1 \)

\(\therefore\ \text{Given}\ \ -1 \leqslant a<b \leqslant 1\ \ \Rightarrow \sin ^{-1} a<\sin ^{-1} b \)

\(\Rightarrow B\)

Trigonometry, EXT1 T1 2023 HSC 5 MC

Which of the following is the value of   \(\sin ^{-1}(\sin a)\)  given that  \(\pi<a<\dfrac{3 \pi}{2}\)? 

  1. \(a-\pi\)
  2. \(\pi-a\)
  3. \(a\)
  4. \(-a\)
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\(B\)

Show Worked Solution

\(\sin(a)\) \(=-\sin(a-\pi)\ \ \text{(see above)}\)  
\(\sin^{-1}(\sin(a))\) \(=\sin^{-1}(-\sin(a-\pi))\)  
  \(=\sin^{-1}(\sin(\pi-a)) \)  
  \(=\pi-a \)  

 
\(\Rightarrow B\)

Mean mark 56%.

Trigonometry, EXT1 T1 EQ-Bank 27

Sketch  `y=3cos^(-1)(2x-1)`   (3 marks)

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`text{Domain:}`

`-1` `<=(2x-1)<=1`  
`0` `<=2x<=2`  
`0` `<=x<=1`  

 
`text{Range of}\ \ cos^(-1)(x)=[0,pi]`

`=>\ text{Range of}\ \ 3cos^(-1)(x)=[0,3pi]`

`text{At}\ \ x=0,\ \ y=3cos^(-1)(-1)=3pi`

`text{Sketch}\ \ y=3cos^(-1)(2x-1):`
 

Trigonometry, EXT1 T1 2022 HSC 13c

The function `f` is defined by  `f(x)=\sin (x)`  for all real numbers `x`. Let `g` be the function defined on  `[-1,1]`  by  `g(x)=\arcsin (x)`.

Is `g` the inverse of `f`? Justify your answer.   (2 marks)

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`text{The domain of}\ \ f(x) = RR`

`text{The range of}\ \ g(x) = [-pi/2,pi/2]`

`text{S}text{ince the domain}\ \ f(x) !=\ text{range of}\ \ g(x),`

`f^(-1)(x)!=g(x)`

Show Worked Solution

`text{The domain of}\ \ f(x) = RR`

`text{The range of}\ \ g(x) = [-pi/2,pi/2]`

`text{S}text{ince the domain}\ \ f(x) !=\ text{range of}\ \ g(x),`

`f^(-1)(x)!=g(x)`


♦♦ Mean mark 24%.

Trigonometry, EXT1 T1 2022 HSC 1 MC

It is given that  `cos((23 pi)/(12))=(sqrt6+sqrt2)/(4)`.

Which of the following is the value of  `cos^(-1)((sqrt6+sqrt2)/(4))`?

  1. `{23pi}/{12}`
  2. `{11pi}/{12}`
  3. `pi/12`
  4. `- {11pi}/{12}`
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`C`

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`(23pi)/12\ \ =>\ \ text{4th quadrant}`

`cos((23pi)/12)=cos(2pi-(23pi)/12)=cos(pi/12)`

`:. cos^(-1)((sqrt6+sqrt2)/(4))=pi/12`

`=>C`


Mean mark 54%.

Trigonometry, EXT1 T1 2021 HSC 9 MC

Which graph represents the function  `y = sin^(-1) (sin x)`?
 

 

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`A`

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`text(By elimination:)`

`text(At)\ \ x = pi, \ y = sin^(-1)(sin pi) = sin^(-1) 0 = 0`

`->\ text(Eliminate C and D)`

`text(At)\ \ x = (2pi)/3, \ y` `= sin^(-1)(sin\ (2pi)/3)`
  `= sin^(-1)(sqrt3/2)`
  `= pi/3`

 
`->\ text(Eliminate B)`

`=>\ A`

Trigonometry, EXT1 T1 EQ-Bank 6 MC

The graph of the function  `y = sin^(-1)(x-2)`  is transformed by being dilated by a factor of 3 from the `y`-axis and then translated to the right by 2.

What is the equation of the transformed graph?

  1. `y = sin^(-1)((x-8)/3)`
  2. `y = sin^(-1)((x-12)/3)`
  3. `y = sin^(-1)(3x-4)`
  4. `y = sin^(-1)(3x-8)`
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`A`

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`y = sin^(-1)(x-2)`
 

`text(Dilate by factor 3 from the)\ ytext(-axis)`

`text(Swap:)\ \ x -> x/3`

`y_1` `= sin^(-1)(x/3-2)`
  `= sin^(-1)((x-6)/3)`

 

`text(Translate to the right by 2)`

`text(Swap:)\ \ x -> x-2`

`y_2` `= sin^(-1)(((x-2)-6)/3)`
  `= sin^(-1)((x-8)/3)`

 
`=>A`

Trigonometry, EXT1 T1 EQ-Bank 5 MC

The graph of the function  `y = arccos(x-3)`  is transformed by being dilated horizontally with a scale factor of `1/2` and then translated to the left by 1.

What is the equation of the transformed graph?

  1. `y = cos^(-1)((x-5)/2)`
  2. `y = cos^(-1)((x-8)/2)`
  3. `y = cos^(-1)(2x-1)`
  4. `y = cos^(-1)(2x-2)`
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`C`

Show Worked Solution

`y = cos^(-1)(x-3)`

`text(Dilate horizontally with scale factor)\ 1/2`

`text(Swap:)\ \ x -> 2x`

`y_1 = cos^(-1)(2x-3)`

 

`text(Translate to the left by 1)`

`text(Swap:)\ \ x -> x + 1`

`y_2` `= cos^(-1)(2 (x + 1)-3)`
  `= cos^(-1)(2x + 2-3)`
  `= cos^(-1)(2x-1)`

 
`=>C`

Trigonometry, EXT1 T1 EQ-Bank 25

Determine the exact value of

`sin^(-1)(sqrt3/2)-sin^(-1)(-sqrt3/2)-cos^(-1)(-1/2)`.   (3 marks)

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`0`

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`text(Base angle:)\ \ sin\ pi/3 = sqrt3/2`

`sin =>\ text(– ve in 4th quadrant)`

`sin^(-1)(sqrt3/2)` `= pi/3`
`sin^(-1)(-sqrt3/2)` `= -pi/3`

 
`text(Base angle:)\ \ cos\ pi/3 = 1/2`

`cos =>\ text(– ve in 2nd quadrant)`

`cos^(-1)(-1/2)` `= pi-pi/3= (2pi)/3`

 
`:. sin^(-1)(sqrt3/2)-sin^(-1)(-sqrt3/2)-cos^(-1)(-1/2)`

`= pi/3-(-pi/3)-(2pi)/3`

`= 0`

Trigonometry, EXT1 T1 EQ-Bank 23

Find `sin theta` given that  `theta = arccos (12/13) + arctan (3/4)`, and  `0<=theta<=pi/2`.   (3 marks)

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`56/65`

Show Worked Solution
`sin theta` `= sin (cos^(-1) (12/13) + tan^(-1)(3/4))`
  `= sin (cos^(-1) (12/13)) cos (tan^(-1) (3/4)) + sin (tan^(-1)(3/4)) cos (cos^(-1)(12/13))`

 
 

`:. sin theta` `= 5/13 xx 4/5 + 3/5 xx 12/13`
  `= 20/65 + 36/65`
  `= 56/65`

Trigonometry, EXT1 T1 2019 HSC 9 MC

Which graph best represents  `y = cos^(-1) (-sin x)`, for  `-pi/2 <= x <= pi/2`?

A.    B.   
C.    D.   
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`D`

Show Worked Solution
`y` `= cos^(-1)(-sin x)`
  `=cos^(-1)(-cos(pi/2-x))`
  `=cos^(-1)(cos[pi-(pi/2-x)])`
  `= cos^(-1)(cos(x + pi/2))`
  `= pi/2+x`

 
`=>  D`

Trigonometry, EXT1 T1 2017 HSC 11d

Sketch the graph of the function  `y = 2cos^(-1)x`.  (2 marks)

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`y = 2cos^(-1)x`

`=> text(Domain)\ -1 <= x <= 1`

`text(Range)\ \ y = 2cos^(-1)x\ \ text(is)\ \ 0 <= y <= pi`

`=> text(Range)\ \ y = 2cos^(-1)x\ \ text(is)\ \ 0 <= y <= 2pi`
 

Trigonometry, EXT1 T1 2007 HSC 5c

Find the exact values of `x` and `y` which satisfy the simultaneous equations

`sin^(-1)\ x + 1/2\ cos^(-1)\ y = pi/3`   and

`3\ sin^(-1)\ x-1/2\ cos^(-1)\ y = (2pi)/3`.   (3 marks)

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`x = 1/sqrt2, \ \ \ y = sqrt3/2`

Show Worked Solution
`sin^(-1)\ x + 1/2\ cos^(-1)\ y` `= pi/3` `\ \ …\ (1)`
`3\ sin^(-1)\ x-1/2\ cos^(-1)\ y` `= (2pi)/3` `\ \ …\ (2)`

 

MARKER’S COMMENT: The use of ther elimination method here proved much more successful than substitution.

`text(Add)\ \ (1) + (2):`

`4\ sin^(-1)\ x` `= pi`
`sin^(-1)\ x` `= pi/4`
`:.x` `= 1/sqrt2`

 
`text(Substitute)\ \ x = 1/sqrt2\ \ text(into)\ (1):`

`sin^(-1)\ 1/sqrt2 + 1/2\ cos^(-1)\ y` `= pi/3`
`pi/4 + 1/2\ cos^(-1)\ y` `= pi/3`
`1/2\ cos^(-1)\ y` `= pi/12`
`cos^(-1)\ y` `= pi/6`
`:.y` `= sqrt3/2`

Trigonometry, EXT1 T1 2007 HSC 2b

Let  `f(x) = 2 cos^(-1)x`.

  1. Sketch the graph of  `y = f(x)`, indicating clearly the coordinates of the endpoints of the graph.  (2 marks)

  2. State the range of  `f(x)`.  (1 mark)

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a.    `text(See Worked Solutions.)`

b.    `0 ≤ y ≤ 2pi`

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a.    `y= 2\ cos^(-1)x`

`text(Domain:)\ -1 ≤ x ≤ 1`

`text{Range:}\ \0 ≤`  `y/2 ≤ pi`  
`0 ≤` `y ≤ 2pi`  

 

 

b.    `text(Range:)\ \ 0 ≤ y ≤ 2pi`

Calculus, EXT1 C2 2006 HSC 2a

Let  `f(x) = sin^-1 (x + 5).`

  1. State the domain and range of the function  `f(x).`  (2 marks)

  2. Find the gradient of the graph of  `y = f(x)`  at the point where  `x = -5.`  (2 marks)

  3. Sketch the graph of  `y = f(x).`  (2 marks)

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a.    `text(Domain):\ -6 <= x <= -4, \ \ \ text(Range):\ -pi/2 <= y <= pi/2`

b.    `1`

c.     
   

Show Worked Solution

a.    `f(x) = sin^-1 (x + 5)`

`text(Domain)`

`-1 <= x + 5 <= 1`

`-6 <= x <= -4`

`text(Range)`

`-pi/2 <= y <= pi/2`

 

b.    `y = sin^-1 (x + 5)`

`(dy)/(dx) = 1/sqrt(1-(x + 5)^2)`
 

`text(When)\ \ x = -5`

`(dy)/(dx)` `= 1/sqrt(1-(-5 + 5)^2)`
  `= 1/sqrt(1-0)`
  `= 1`

 
`:.\ text(Gradient of)\ \ y = f(x)\ \ text(at)\ \ x = -5\ \ text(is)\ \ 1.`

 

c.    

 EXT1 2006 2a

Trigonometry, EXT1 T1 2005 HSC 1c

State the domain and range of  `y = cos^-1 (x/4).`   (2 marks)

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`text(Domain)\ \ -4 <= x <= 4`

`text(Range)\ \ 0 <= y <= pi`

Show Worked Solution

 `y = cos^-1\ x/4`

`text(Domain of)\ \ y =cos^-1 x\ \ text(is)`

`-1 <= x <= 1`

`:.\ text(Domain of)\ \ y = cos^-1\ x/4\ \ text(is)`

`-1 <= x/4 <= 1`

`-4 <= x <= 4`

 

`text(Range)\  \y = cos^-1\ x\ \ text(is)`

`0 <= y <= pi`

`:.\ text(Range)\ \ y = cos^-1\ x/4\ \ text(is)`

`0 <= y <= pi`

Trigonometry, EXT1 T1 2015 HSC 6 MC

What is the domain of the function `f(x) = sin^(-1)\ (2x)`?

  1. `-pi ≤ x ≤ pi`
  2. `-2 ≤ x ≤ 2`
  3. `-pi/4 ≤ x ≤ pi/4`
  4. `-1/2 ≤ x ≤ 1/2`
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`D`

Show Worked Solution

`f(x)= sin^(-1)\ (2x)`

`text(Domain of)\ f(x) = sin^(-1) x\ \ text(is)`

`-1 ≤ x ≤ 1`

`:.\ text(Domain of)\ \ f(x) = sin^(-1)\ (2x)\ \ text(is)`

`-1 ≤ 2x ≤ 1`

`-1/2 ≤ x ≤ 1/2`

`=> D`

Trigonometry, EXT1 T1 2013 HSC 9 MC

The diagram shows the graph of a function. 
 

2013 9 mc
 

 Which function does the graph represent? 

  1. `y = cos^(-1) x`
  2. `y = pi/2 + sin^(-1) x`
  3. `y = - cos^(-1) x`
  4. `y = - pi/2\ - sin^(-1) x`
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`B`

Show Worked Solution

`text(By elimination,)`

`text(The graph passes through)\ \ (1, pi)`

`text(The only equation to satisfy this point is)`

`y = pi/2 + sin^(-1) x`

`=>  B`

Trigonometry, EXT1 T1 2011 HSC 2d

Sketch the graph of the function  `f(x) = 2arccos x`.  Clearly indicate the domain and range of the function.    (2 marks)

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Show Worked Solution
`y` `= 2 cos^(-1) x`
`y/2` `= cos^(-1)x`

`text(Domain)\ -1 <= x <= 1`

`text(S)text(ince)\ \ 0 <= y/2 <= pi`

`text(Range)\ 0 <= y <= 2pi`

EXT1 2011 2d

Trigonometry, EXT1 T1 2012 HSC 13a

Write  `sin(2 cos ^(-1) (2/3))`  in the form  `a sqrtb`, where `a` and `b` are rational.   (2 mark)

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 `4/9 sqrt5` 

Show Worked Solution
TIP: This question is made  less complicated by reminding yourself that `cos^(-1) (2/3)` is simply an angle.

`sin (2 cos^(-1) (2/3))`

`text(Let)\ \ x = cos ^(-1) (2/3)`

`:.\ cos x = 2/3`
 

Inverse Functions, EXT1 2012 HSC 13a Answer 
 

`sin x = sqrt5/3`

`sin(2 cos^(-1) (2/3))` `= sin 2x`
  `= 2 sin x cos x`
  `= 2 xx sqrt5/3 xx 2/3`
  `= 4/9 sqrt5`

Trigonometry, EXT1 T1 2012 HSC 4 MC

Which function best describes the following graph? 

2012 4 mc

  1. `y = 3sin^(−1) 2x` 
  2. `y = 3/2 sin^(−1) 2x` 
  3. `y = 3sin^(−1)\ x/2` 
  4. `y = 3/2 sin^(−1)\ x/2` 
Show Answers Only

`C`

Show Worked Solution
`text{Domain:}` `\ \ -2 <= x <= 2`
  `\ \ -1 <= x/2 <= 1`

 
`:.\ text(Graph is)\ \ y = a sin^(-1)\ x/2`
 

`text(When)\ \ x = 2,\ \ y = (3pi)/2:`

`(3pi)/2` `=a sin^(-1) 1`
`(3pi)/2` `=a xx pi/2`
`a` `= 3`
`:.\ y` `= 3 sin ^(-1)\ x/2`

 
`=>  C`