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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 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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\(A\)

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

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

\(\Rightarrow A\)

Trigonometry, EXT1 T1 EQ-Bank 4

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 3 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`

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`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 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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  1. `text(See Worked Solutions.)`
  2. `0 ≤ y ≤ 2pi`
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i.   `y= 2\ cos^(-1)x`

`text(Domain:)\ -1 ≤ x ≤ 1`

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

 

 

ii.  `text(Range:)\ \ 0 ≤ y ≤ 2pi`