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Trigonometry, EXT1 T2 EQ-Bank 8

  1. Show that  \(\dfrac{\sin 2 x}{1+\cos 2 x}=\tan x\).  (1 mark)

  2. Hence find the exact value of \(\tan \dfrac{\pi}{8}\) in simplest form.  (2 marks)

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a.   \(\dfrac{\sin 2 x}{1+\cos 2 x}\) \(=\dfrac{2\sin x \cos x}{2\cos^{2} x}\)  
  \(=\dfrac{\sin x}{\cos x}\)  
  \(=\tan x\)  

b.   \(\sqrt2-1\)

Show Worked Solution

a.   \(\dfrac{\sin 2 x}{1+\cos 2 x}\) \(=\dfrac{2\sin x\, \cos x}{2\cos^{2} x}\)  
  \(=\dfrac{\sin x}{\cos x}\)  
  \(=\tan x\)  

 

b.    \(\tan \dfrac{\pi}{8}\) \(=\dfrac{\sin \frac{\pi}{4}}{1+\cos \frac{\pi}{4}}\)
    \(= \dfrac{\frac{1}{\sqrt2}}{1+\frac{1}{\sqrt2}} \times \dfrac{\sqrt2}{\sqrt2} \)
    \(= \dfrac{1}{\sqrt2+1} \times \dfrac{\sqrt2-1}{\sqrt2-1}\)
    \(=\sqrt2-1\)

Trigonometry, EXT1 T2 2013 HSC 8 MC

The angle  `theta`  satisfies  `sin theta = 5/13`  and  `pi/2 < theta < pi`.

What is the value of  `sin 2 theta`?

  1. `10/13`
  2. `- 10/13` 
  3. `120/169`
  4. `- 120/169`
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`D`

Show Worked Solution
 
♦ Mean mark 44% 

`sin theta = 5/13\ \ \ (pi/2 < theta < pi)`

`text(S)text(ince)\ \ theta\ \ text(is in the 2nd quadrant,)`

`cos theta = -12/13`

`:.sin 2theta` `= 2 sin theta cos theta`
  `= 2 xx 5/13 xx -12/13`
  `= -120/169`

`=>  D`