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Trigonometry, 2ADV T2 EQ-Bank 1 MC

Determine the number of values of \(\theta\) in the range  \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\)  that satisfy the equation

\((\tan \theta-\sqrt{3})(\cos^{2}\theta-1)=0 \)

  1. \(3\)
  2. \(4\)
  3. \(5\)
  4. \(6\)
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\(C\)

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\(\tan \theta = \sqrt{3}\ \rightarrow \text{2 solutions} \)

\(\cos^{2}\theta\) \(=1\)  
\(\cos\theta\) \(= \pm 1\)  
\(\theta\) \(=0^{\circ}, 180^{\circ}, 360^{\circ}\ \rightarrow \text{3 solutions} \)  

 
\(\Rightarrow C\)

Trigonometry, 2ADV T2 SM-Bank 38

Solve  `2cos(2x) = −sqrt3`  for  `x`, where  `0 <= x <= pi`.  (2 marks) 

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`x = (5pi)/12, (7pi)/12`

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`cos(2x)` `= – sqrt3/2`
`2x` `= (5pi)/6, 2pi – (5pi)/6, 2pi+(5pi)/6`
  `=(5pi)/6, (7pi)/6, (17pi)/6,\ …`
`:. x` `=(5pi)/12, (7pi)/12\ \ \ (0 <= x <= pi)`

Trigonometry, 2ADV T2 SM-Bank 37

Solve the equation  `cos((3x)/2) = 1/2`  for  `−pi/2<=x<=pi/2`.  (2 marks)

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`x = ± (2pi)/9`

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`cos((3x)/2) = 1/2`

`=>\ text(Base angle)\ =pi/3`

`(3x)/2` `= (-pi)/3, pi/3, (5pi)/3, …`
`:. x` `=(-2pi)/9, (2pi)/9, (10pi)/9`
  `= (-2pi)/9, (2pi)/9\ \ \ (−pi/2<=x<=pi/2)`