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Graphs, SPEC2 2019 VCAA 3 MC

The implied domain of the function with rule  `f(x) = 1 - sec(x + pi/4)`  is

  1. `R`
  2. `[0,2]`
  3. `R\ text(\)\ {((4n - 1)pi)/4}, n ∈ ZZ`
  4. `R\ text(\)\ {((4n + 1)pi)/4}, n ∈ ZZ`
  5. `R\ text(\)\ {((2n - 1)pi)/4}, n ∈ ZZ`
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`D`

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`y = sec (x)=1/cos(x)\ \ text(has asymptotes when)`

`x = −pi/2, pi/2, (3pi)/2, …`

`=> y = sec (x + pi/4)\ \ text(has asymptotes at when)`

`x = (−3pi)/4, pi/4, (5pi)/4, …`

`:.\ text(Domain:)\ R\ text(\)\ {((4n + 1)pi)/4}, n ∈ ZZ`

`=>D`

Trigonometry, SPEC1 2011 VCAA 8

Find the coordinates of the points of intersection of the graph of the relation

`y = text(cosec)^2 ((pi x)/6)`  with the line  `y = 4/3`, for  `0 < x < 12.`  (3 marks)

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`(2, 4/3),(4, 4/3),(8, 4/3), (10, 4/3)`

Show Worked Solution

`text(Intersection occurs when:)`

♦ Mean mark 46%.

`text(cosec)^2((pix)/6)` `=4/3`
`text(cosec)((pix)/6)` `= ±2/sqrt3`
`sin((pix)/6)` `= ±sqrt3/2`

 
`text(Given:)\ \ 0 < x < 12 \ \ =>\ \ 0 < (pix)/6 < 2pi`
 

`(pix)/6` `= pi/3, pi – pi/3, pi + pi/3, 2pi – pi/3`
  `= pi/3, (2pi)/3, (4pi)/3,(5pi)/3`
`x` `= 2, 4, 8, 10`

  
`=> y = 4/3\ text(for each)`

`:.\ text(Intersection at:)\ \ (2, 4/3),(4, 4/3),(8, 4/3), (10, 4/3)`