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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 5

  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 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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\(B\)

Show Worked Solution

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

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

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

\(\Rightarrow B\)

Trigonometry, EXT1 T1 2021 HSC 9 MC

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

 

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`A`

Show Worked Solution

`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 4 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`

Show Worked Solution

`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`

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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  1. `text(Domain):\ -6 <= x <= -4, \ \ \ text(Range):\ -pi/2 <= y <= pi/2`
  2. `1`
  3.  

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i.  `f(x) = sin^-1 (x + 5)`

`text(Domain)`

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

`-6 <= x <= -4`

`text(Range)`

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

 

ii.  `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.`

 

iii.

 EXT1 2006 2a

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`

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`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 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` 
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`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`