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Advanced Trigonometry, SMB-042

Find all the values of \(\theta\), where  \(-180^{\circ} \leq \theta \leq 180^{\circ}\), such that

\(\tan\,\theta(\tan\,\theta-1)=0\)   (3 marks)

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\(\theta=-180^{\circ}, -135^{\circ}, 0^{\circ}, 45^{\circ}, 180^{\circ}\)

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\(\text{If}\ \ \tan\,\theta=0\ \ \Rightarrow\ \ \theta=0^{\circ}, 180^{\circ}, -180^{\circ} \)

\(\text{If}\ \ \tan\,\theta-1=0\ \ \Rightarrow\ \ \tan\,\theta=1\)

\(\text{Reference angle:}\ \ \tan\,\theta=1\ \ \Rightarrow \ \ \theta = 45^{\circ}\)

\(\text{tan is positive in 1st/3rd quadrants.}\)

\(\theta=45^{\circ}, (180+45)^{\circ}=45^{\circ}, 225^{\circ} = 45^{\circ}, -135^{\circ}\ \ (-180^{\circ} \leq \theta \leq 180^{\circ}) \)

\(\therefore \theta=-180^{\circ}, -135^{\circ}, 0^{\circ}, 45^{\circ}, 180^{\circ}\)

Advanced Trigonometry, SMB-041

Identify all \(\theta\) values between  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

\(\tan\,\theta +1=0\)   (2 marks)

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\(\theta=135^{\circ}, 315^{\circ}\)

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\(\tan\,\theta +1=0\ \ \Rightarrow \ \ \tan\,\theta=-1\)

\(\text{Reference angle:}\ \ \tan\,\theta=1\ \ \Rightarrow \ \ \theta = 45^{\circ}\)

\(\text{tan is negative in 2nd/4th quadrants.}\)

\(\theta=(180-45)^{\circ}, (360-45)^{\circ}=135^{\circ}, 315^{\circ}\)

Advanced Trigonometry, SMB-040

Identify all \(\theta\) values between  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

\(\tan^{2}\theta=\dfrac{1}{3} \)   (2 marks)

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\(\theta=30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}\)

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\(\tan^{2}\theta\) \(=\dfrac{1}{3} \)  
\(\tan\,\theta\) \(=\pm \dfrac{1}{\sqrt{3}} \)  

 
\(\text{Reference angle:}\ \ \tan\,\theta=\dfrac{1}{\sqrt{3}}\ \ \Rightarrow \ \ \theta = 30^{\circ}\)

\(\text{Angles exist in all quadrants:}\)

\(\theta\) \(=30^{\circ}, (180-30)^{\circ},(180+30)^{\circ},(360-30)^{\circ}\)  
  \(=30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}\)  

Advanced Trigonometry, SMB-039

Find every angle, \(\theta\), between  \(0^{\circ} \leq \theta \leq 360^{\circ}\), for which

\(\tan\,\theta=-\sqrt{3}\)   (2 marks)

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\(\theta=120^{\circ}, 300^{\circ}\)

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\(\text{Reference angle:}\ \ \tan\,\theta=-\sqrt{3}\ \ \Rightarrow \ \ \theta = 60^{\circ}\)

\(\text{tan is negative in 2nd/4th quadrants.}\)

\(\theta=(180-60)^{\circ}, (360-60)^{\circ}=120^{\circ}, 300^{\circ}\)

Advanced Trigonometry, SMB-038

Identify all \(\theta\) values between  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

\(\tan\,\theta=-\dfrac{1}{\sqrt{3}} \)   (2 marks)

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\(\theta=150^{\circ}, 330^{\circ}\)

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\(\text{Reference angle:}\ \ \tan\,\theta=\dfrac{1}{\sqrt{3}}\ \ \Rightarrow \ \ \theta = 30^{\circ}\)

\(\text{tan is negative in 2nd/4th quadrants.}\)

\(\theta=(180-30)^{\circ}, (360-30)^{\circ}=150^{\circ}, 330^{\circ}\)

Advanced Trigonometry, SMB-037

Identify all \(\theta\) values between  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

\(\tan\,\theta=\sqrt{3} \)   (2 marks)

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\(\theta=60^{\circ}, 240^{\circ}\)

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\(\text{Reference angle:}\ \ \tan\,\theta=\sqrt{3}\ \ \Rightarrow \ \ \theta = 60^{\circ}\)

\(\text{tan is positive in 1st/3rd quadrants.}\)

\(\theta=60^{\circ}, (180+60)^{\circ}=60^{\circ}, 240^{\circ}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 43v1

Find the exact value of

\(\tan(-150^{\circ})\).   (2 marks)

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\(\dfrac{1}{\sqrt{3}}\)

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\(\tan(-150^{\circ})= \tan\,210^{\circ}\)

\(\text{Reference angle:}\ 180+\theta=210\ \ \Rightarrow \ \ \theta=30^{\circ}\)

\(\text{Since tan is positive in 3rd quadrant:}\)

\(\tan(-150^{\circ})= \tan\,30^{\circ}=\dfrac{1}{\sqrt{3}}\)

Advanced Trigonometry, 2ADV T2 2012 HSC 6 MC

What are the solutions of  `sqrt3 tanx = -1`  for  `0^@<=x<=360^@`? 

  1. `120^@\ text(and)\ 240^@` 
  2. `120^@\ text(and)\ 300^@`
  3. `150^@\ text(and)\ 210^@` 
  4. `150^@\ text(and)\ 330^@` 
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`D`

Show Worked Solution
`sqrt3 tanx` `= -1`
`tanx` `= -1/sqrt3`

 
`text(When)\ tanx = 1/sqrt3,\ \ x=30^@`

`text(S)text(ince)\ tanx\ text{is negative in 2nd/4th quadrant:}`

`:. x` ` = 180-30,360-30`
  `= 150^@,\ 330^@`

 
`=>  D`

Advanced Trigonometry, SMB-010

  1. Determine the reference angle for \(210^{\circ}\).   (1 mark)

  2. Using part (a) or otherwise, give the exact value for \(\tan 210^{\circ}\)   (2 marks)

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a.   \(30^{\circ}\)

b.   \(\dfrac{1}{\sqrt{3}}\)

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a.   \(210^{\circ}\ \text{is in the 3rd quadrant.}\)

\(\text{Reference angle:}\ \ 180+\theta=210^{\circ}\ \ \Rightarrow \ \ \theta = 30^{\circ}\)
  

b.   \(\tan 30^{\circ} = \dfrac{1}{\sqrt{3}}\)

\(\Rightarrow \ \tan \theta>0\ \ \text{in 3rd quadrant}\)

\(\therefore \tan 210^{\circ} = \dfrac{1}{\sqrt{3}}\)

Advanced Trigonometry, SMB-009

  1. Determine the reference angle for \(120^{\circ}\).   (1 mark)

  2. Using part (a) or otherwise, give the exact value for \(\tan 120^{\circ}\)   (2 marks)

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a.   \(60^{\circ}\)

b.   \(-\sqrt{3}\)

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a.   \(120^{\circ}\ \text{is in the 2nd quadrant.}\)

\(\text{Reference angle:}\ \ 180-120=60^{\circ}\)
  

b.   \(\tan 60^{\circ} = \sqrt{3}\)

\(\Rightarrow \ \tan \theta<0\ \ \text{in 2nd quadrant}\)

\(\therefore \tan 120^{\circ} = -\sqrt{3}\)