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Trigonometry, 2ADV T3 EQ-Bank 6

  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)

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a.    
           

b.   \(x=\dfrac{2\pi}{3}\)

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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 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)

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`pi/24, (5pi)/24`

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`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)`