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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)

Show Answers Only

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

Show Worked Solution

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

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

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

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

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

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

Advanced Trigonometry, SMB-041

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

\(\tan\,\theta +1=0\)   (2 marks)

Show Answers Only

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

Show Worked Solution

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

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

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

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

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)

Show Answers Only

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

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

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

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

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

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

Show Worked Solution

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

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

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

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

Show Worked Solution

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

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

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

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

Show Worked Solution

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

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

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

Advanced Trigonometry, SMB-034

Find every angle, \(\theta\), between  \(-180^{\circ} \leq \theta \leq 180^{\circ}\), for which

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

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\(\theta=30^{\circ}, -30^{\circ}\)

Show Worked Solution

\(\text{Reference angle:}\ \ \cos\,\theta=\dfrac{\sqrt{3}}{2}\ \ \Rightarrow \ \ \theta = 30^{\circ}\)

\(\text{cos is positive in 1st/4th quadrants.}\)

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

\(\therefore \theta=30^{\circ}, -30^{\circ}\ \ \ (-180^{\circ} \leq \theta \leq 180^{\circ}) \)

Advanced Trigonometry, SMB-036

Find all the values of \(\theta\), where  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

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

Show Answers Only

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

Show Worked Solution

\(\text{Reference angle:}\ \ \cos\,\theta=\dfrac{\sqrt{3}}{2}\ \ \Rightarrow \ \ \theta = 30^{\circ}\)

\(\text{cos is positive in 1st/4th quadrants.}\)

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

Advanced Trigonometry, SMB-035

Solve for all \(\theta\) in the range  \(0^{\circ} \leq \theta \leq 360^{\circ}\), that make the following equation correct

\(\cos^{2}\theta-\cos\,\theta=0\)   (3 marks)

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\(\theta=0^{\circ},90^{\circ},270^{\circ},360^{\circ}\)

Show Worked Solution
\(\cos^{2}\theta-\cos\,\theta\) \(=0\)  
\(\cos\,\theta(\cos\,\theta-1)\) \(=0\)  

 
\(\text{If}\ \ \cos\,\theta=0\ \ \Rightarrow\ \ \theta=90^{\circ}, 270^{\circ}\)

\(\text{If}\ \ \cos\,\theta-1=0\ \ \Rightarrow\ \ \cos\,\theta=1\ \ \Rightarrow\ \ \theta=0^{\circ}, 360^{\circ}\)

\(\theta=0^{\circ},90^{\circ},270^{\circ},360^{\circ}\)

Advanced Trigonometry, SMB-033

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

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

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\(\theta=135^{\circ}, 225^{\circ}\)

Show Worked Solution

\(\text{Reference angle:}\ \ \cos\,\theta=\dfrac{1}{\sqrt{2}}\ \ \Rightarrow \ \ \theta = 45^{\circ}\)

\(\text{cos is negative in 2nd/3rd quadrants.}\)

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

Advanced Trigonometry, SMB-032

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

\(\cos\,\theta=\dfrac{1}{2}\)   (2 marks)

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\(\theta=60^{\circ}, 300^{\circ}\)

Show Worked Solution

\(\text{Reference angle:}\ \ \cos\,\theta=\dfrac{1}{2}\ \ \Rightarrow \ \ \theta = 60^{\circ}\)

\(\text{cos is positive in 1st/4th quadrants.}\)

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

Advanced Trigonometry, SMB-031

Find all the values of \(\theta\), where  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

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

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\(\theta=60^{\circ}, 120^{\circ}\)

Show Worked Solution

\(\text{Reference angle:}\ \ \sin\,\theta=\dfrac{\sqrt{3}}{2}\ \ \Rightarrow \ \ \theta = 60^{\circ}\)

\(\text{sin is positive in 1st/2nd quadrants.}\)

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

Advanced Trigonometry, SMB-030

Solve for all \(\theta\) in the range  \(0^{\circ} \leq \theta \leq 360^{\circ}\), that make the following equation correct

\(\sin\,\theta(\sin\,\theta+1)=0\)   (3 marks)

Show Answers Only

\(\theta=0^{\circ},180^{\circ},270^{\circ},360^{\circ}\)

Show Worked Solution

\(\text{If}\ \ \sin\,\theta=0\ \ \Rightarrow\ \ \theta=0^{\circ}, 180^{\circ}, 360^{\circ}\)

\(\text{If}\ \ \sin\,\theta+1=0\ \ \Rightarrow\ \ \sin\,\theta=-1\ \ \Rightarrow\ \ \theta=270^{\circ}\)

\(\theta=0^{\circ},180^{\circ},270^{\circ},360^{\circ}\)

Advanced Trigonometry, SMB-029

Find all the values of \(\theta\), where  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

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

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\(\theta=225^{\circ}, 315^{\circ}\)

Show Worked Solution

\(\text{Reference angle:}\ \ \sin\,\theta=\dfrac{1}{\sqrt{2}}\ \ \Rightarrow \ \ \theta = 45^{\circ}\)

\(\text{sin is negative in 3rd/4th quadrants.}\)

\(\theta=(180+45)^{\circ}, (360-45)^{\circ}=225^{\circ}, 315^{\circ}\)

Advanced Trigonometry, SMB-028

Find all the values of \(\theta\), where  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

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

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\(\theta=240^{\circ}, 300^{\circ}\)

Show Worked Solution

\(\text{Reference angle:}\ \ \sin\,\theta=\dfrac{\sqrt{3}}{2}\ \ \Rightarrow \ \ \theta = 60^{\circ}\)

\(\text{sin is negative in 3rd/4th quadrants.}\)

\(\theta=(180+60)^{\circ}, (360-60)^{\circ}=240^{\circ}, 300^{\circ}\)

Advanced Trigonometry, SMB-027

Find all the values of \(\theta\), where  \(0^{\circ} \leq \theta \leq 360^{\circ}\), such that

\(\sin\,\theta=\dfrac{1}{2}\)   (2 marks)

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

Show Worked Solution

\(\text{Reference angle:}\ \ \sin\,\theta=\dfrac{1}{2}\ \ \Rightarrow \ \ \theta = 30^{\circ}\)

\(\text{sin is positive in 1st/2nd quadrants.}\)

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

Advanced Trigonometry, 2ADV T2 SM-Bank 40

Let  \((\tan\,\theta-1) (\sin\,\theta-\sqrt{3}\cos\,\theta)(\sin\,\theta + \sqrt{3}\cos\,\theta) = 0\).

  1. State all possible values of \(\tan\,\theta\).   (1 mark)

  2. Hence, find all possible solutions for \((\tan\,\theta-1) (\sin^{2}\theta-3\cos^{2}\theta) = 0\), where \(\ \ 0^{\circ} \leq \theta \leq 180^{\circ}\)   (2 marks)

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a.   \(\tan\,\theta = 1\ \ \text{or}\ \ \tan\,\theta = \pm \sqrt {3}\)

b.   \(\theta = 45^{\circ},60^{\circ},120^{\circ}\)

Show Worked Solution

a.   \((\tan\,\theta-1) (\sin\,\theta-\sqrt{3}\cos\,\theta)(\sin\,\theta + \sqrt{3}\cos\,\theta) = 0\)

\(\Rightarrow \tan\,\theta = 1\)

♦ Mean mark 42%.
\(\Rightarrow \sin\,\theta-\sqrt{3}\cos\,\theta\) \(=0\)
\(\sin\,\theta\) \(=\sqrt{3}\cos\,\theta\)
\(\tan\,\theta\) \(=\sqrt{3}\)

 

\(\Rightarrow\sin\,\theta + \sqrt{3}\cos\,\theta\) \(=0\)
\(\sin\,\theta\) \(=-\sqrt{3}\cos\,\theta\)
\(\tan\,\theta\) \(=-\sqrt{3}\)

 
\(\therefore \tan\,\theta = 1\ \ \text{or}\ \ \tan\,\theta = \pm \sqrt{3}\)
  

b.   \((\tan\,\theta-1)(\sin^2\theta-3\cos^2\theta) = 0\)

\(\text{Using part a:}\)

♦ Mean mark 42%.

\((\tan\,\theta-1) (\sin\,\theta-\sqrt{3}\cos\,\theta)(\sin\,\theta + \sqrt{3}\cos\,\theta) = 0\)

\(\Rightarrow \tan\,\theta\) \(= 1\) \(\ \ \ \ \ \ \ \ \ \) \(\tan\,\theta\) \(= \pm \sqrt{3}\)
\(\theta\) \(= 45^{\circ}\)   \(\theta\) \(= 60^{\circ}, 120^{\circ}\)

 
\(\therefore \theta = 45^{\circ},60^{\circ},120^{\circ}\ \ \ \ (0^{\circ} \leq \theta \leq 180^{\circ})\)

Advanced Trigonometry, 2ADV T2 2019 HSC 13a

Solve  \(2\sin\,\theta\,\cos\,\theta=\sin\,\theta\)  for  \(0^{\circ} \leq \theta \leq 360^{\circ}\).   (3 marks)

Show Answers Only

\(\theta=0^{\circ}, 60^{\circ}, 180^{\circ}, 300^{\circ}, 360^{\circ}\)

Show Worked Solution

♦ Mean mark 49%.

\(2\sin\,\theta\,\cos\,\theta\) \(=\sin\,\theta\)
\(2\sin\,\theta\,\cos\,\theta-\sin\,\theta\) \(= 0\)
\(\sin\,\theta(2\cos\,\theta-1)\) \(=0\)

 
\(\sin\,\theta=0:\)

\(\Rightarrow \theta=0^{\circ}, 180^{\circ}, 360^{\circ}\)
 

\(\cos\,\theta=\dfrac{1}{2}:\)

\(\text{Reference angle}\ =60^{\circ}\)

\(\text{cos is positive in 1st/4th quadrants.}\)

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

Advanced Trigonometry, 2ADV T2 2004 HSC 8a

  1. Show that  \(\cos\,\theta\,\tan\,\theta = \sin\,\theta\).   (1 mark)

  2. Hence solve  \(8\sin\,\theta\,\cos\,\theta\,\tan\,\theta = \dfrac{1}{\sin\,\theta}\)  for  \(0^{\circ} \leq \theta \leq 360^{\circ}\).   (2 marks)

Show Answers Only
  1. \(\text{Proof (See Worked Solutions)}\)
  2. \(30^{\circ}, 150^{\circ}\)
Show Worked Solution

a.    \(\text{Prove}\ \cos\,\theta\,\tan\,\theta = \sin\,\theta\)

\(\text{LHS}\) \(= \cos\,\theta\,\tan\,\theta\)
  \(=\cos\,\theta \times \dfrac{\sin\,\theta}{\cos\,\theta}\)
  \(= \sin\,\theta\)
  \(=\ \text{RHS}\)

 

b.    \(8\sin\,\theta\,\cos\,\theta\,\tan\,\theta\) \(= \dfrac{1}{\sin\,\theta}\)
  \(8\sin^{2}\theta\) \(= \dfrac{1}{\sin\,\theta}\)
  \(8\sin^{3}\theta\) \(= 1\) 
  \(\sin^{3}\theta\) \(=\dfrac{1}{8}\)
  \(\sin\,\theta\)  \(=\dfrac{1}{2}\) 

 
\(\text{Reference angle:}\ \theta=30^{\circ}\)

\(\text{sin is positive in 1st/2nd quadrants.}\)

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

Advanced Trigonometry, 2ADV T2 SM-Bank 34

Solve the equation  \(\sqrt{3}\sin\,\theta = \cos\,\theta\)  for  \(-180^{\circ} \leq \theta \leq 180^{\circ}\).   (2 marks)

Show Answers Only

\(\theta = 30^{\circ}\ \text{or}\ -150^{\circ}\)

Show Worked Solution

\(\text{Divide both sides by}\ \cos\,\theta:\)

MARKER’S COMMENT: Many students who found the base angle correctly could not solve within the restrictions.
\(\sqrt{3}\sin\,\theta\) \(=\cos\,\theta\)
\(\sqrt{3}\tan\,\theta\) \(= 1\)
\(\tan\,\theta\) \(= \dfrac {1}{\sqrt{3}}\)

 
\(\text{Reference angle:}\ \theta=30^{\circ}\)

\(\therefore\ \theta = 30^{\circ}\ \text{or}\ -150^{\circ}, \  \text{for }-180^{\circ} \leq \theta \leq 180^{\circ}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 2

Find all solutions of the equation  \(2 \cos\,\theta = \dfrac{\sqrt{3}}{\tan\,\theta}\),  for  \(0^{\circ} \leq \theta \leq 360^{\circ}\)   (3 marks)

Show Answers Only

\(\theta = 60^{\circ}, 90^{\circ}, 120^{\circ}, 270^{\circ}\)

Show Worked Solution

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

\(2 \cos\,\theta-\dfrac{\sqrt{3}\cos\,\theta}{\sin\,\theta}\) \(= 0\)  
\(\cos\,\theta\left(2-\dfrac{\sqrt{3}}{\sin\,\theta}\right) \) \(=0\)  

 
\(\text{If}\ \ \cos\,\theta=0:\)

\(\Rightarrow\ \ \theta=90^{\circ}, 270^{\circ}\)
  

\(\text{If}\ \ 2-\dfrac{\sqrt{3}}{\sin\,\theta} = 0\ \Rightarrow\ \ \sin\,\theta = \dfrac{\sqrt{3}}{2}:\)

\(\Rightarrow\ \ \theta = 60^{\circ}, 120^{\circ}\)

Advanced Trigonometry, 2ADV T2 EQ-Bank 5

Solve  \(\sin x-\cos x=0 \quad-180^{\circ} \leqslant x \leqslant 180^{\circ}\)   (2 marks)

Show Answers Only

\(x=45^{\circ}, -135^{\circ}\)

Show Worked Solution
  \(\sin x-\cos x\) \(=0\)
  \(\dfrac{\sin x}{\cos x}-\dfrac{\cos x}{\cos x}\) \(=0\)
  \(\tan x-1\) \(=0\)
  \(\tan x\) \(=1\)
  \(x\) \(=\tan ^{-1}(1)\)

 
\(\therefore x=45^{\circ}, -135^{\circ}\)

Advanced Trigonometry, 2ADV T2 EQ-Bank 2

Express  \(\dfrac{3}{\sin(180+\theta)}+5 \cos (90-\theta)\)  as a single fraction in terms of \(\sin \theta\), given all angles are measured in degrees.   (3 marks)

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\(\dfrac{-3+5 \sin ^2 \theta}{\sin\,\theta}\)

Show Worked Solution

\(\dfrac{3}{\sin(180+\theta)}+5 \cos (90-\theta)\)

\(=-\dfrac{3}{\sin\,\theta}+5 \sin\,\theta\)

\(=-\dfrac{3}{\sin\,\theta}+\dfrac{5\sin^{2}\theta}{\sin\,\theta}\)

\(=\dfrac{-3+5 \sin ^{2}\theta}{\sin\,\theta}\)

Advanced Trigonometry, 2ADV T2 2014 HSC 7 MC

How many solutions of the equation  `(sin theta-1)(tan theta + 2) = 0`  lie between `0°` and `360°`?

  1. `1`
  2. `2`
  3. `3`
  4. `4`
Show Answers Only

`B`

Show Worked Solution
♦♦♦ Mean mark 25%, making it the toughest MC question in the 2014 exam.

`text(When)\ (sin theta-1)(tan theta + 2) = 0`

`(sin theta-1) = 0\ \ text(or)\ \ tan theta + 2 = 0`

`text(If)\ \ sin theta-1= 0:`

`sin theta= 1\ \ =>\ \ theta= 90°,\ \ \ 0° < theta < 360°`
 

`text(If)\ \ tan theta + 2= 0:`

`tan theta= -2`

`text{Since}\ tan\ 90°\ text{is undefined, there are only 2 solutions when}`

`tan theta = -2\ \text{(which occurs in the 1st and 4th quadrants).}`
  

`:.\ 2\ text(solutions)`

`=>  B`

Advanced Trigonometry, 2ADV T2 2008 HSC 6a

Solve  \(2\sin^{2}\left( \dfrac{\theta}{3}\right) = 1\)  for  \(-180^{\circ} \leq \theta \leq 180^{\circ}\).   (3 marks)

Show Answers Only

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

Show Worked Solution
\(2\sin^{2}\left( \dfrac{\theta}{3}\right)\) \(= 1\)
\(\sin^{2}\left( \dfrac{\theta}{3}\right)\) \(= \dfrac{1}{2}\)
\(\sin\left( \dfrac{\theta}{3}\right)\) \(= \pm \dfrac{1}{\sqrt{2}}\)

 

MARKER’S COMMENT: Many students had problems adjusting their answer to the given domain, especially when dealing with negative angles.

\(\text{When}\ \ \sin\left( \dfrac{\theta}{3}\right)= \dfrac{1}{\sqrt{2}}:\)

\(\dfrac{\theta}{3}\) \(= 45, 180-45=45^{\circ}, 135^{\circ}\)
\(\theta\) \(= 135^{\circ}, 405^{\circ}\)

 

\(\text{When}\ \ \sin\left( \dfrac{\theta}{3}\right)= -\dfrac{1}{\sqrt{2}}:\)

\(\dfrac{\theta}{3}\) \(= -45^{\circ}, -135^{\circ}\)
\(\theta\) \(= -135^{\circ}, -405^{\circ}\)

 

\(\therefore \theta=-135^{\circ}, 135^{\circ}\ \ \text{for}\ \ -180^{\circ} \leq \theta \leq 180^{\circ}\)

Advanced Trigonometry, 2ADV T2 2023 HSC 20v2

Find all the values of `theta`, where  `0^@ <=theta <= 360^@`, such that

`sin(theta-30^@)=-1/2`   (3 marks)

Show Answers Only

`theta=0^@, 240^@ and 360^@`

Show Worked Solution

`sin30^@=1/2\ \ =>\ \ text{Reference angle}\ =30^@`

`=>\ text{sin is negative in 3rd and 4th quadrants}`

`sin(theta-30^@)` `=180+30, 360-30`  
  `=210^@, 330^@`  

 
`theta-30^@=210^@\ \ =>\ \ theta=240^@`

`theta-30^@=330^@\ \ =>\ \ theta=360^@`

 
`text{Consider}\ theta = 0^@`

`sin(0-30^@)=sin(-30^@)=-1/2`

`:.theta=0^@, 240^@ and 360^@`

Advanced Trigonometry, 2ADV T2 2023 HSC 20

Find all the values of `theta`, where  `0^@ <=theta <= 360^@`, such that

`sin(theta-60^@)=-sqrt3/2`   (3 marks)

Show Answers Only

`theta=0^@, 300^@ and 360^@`

Show Worked Solution

`sin60^@=sqrt3/2\ \ =>\ \ text{Reference angle}\ =60^@`

`=>\ text{sin is negative in 3rd and 4th quadrants}`

`sin(theta-60^@)` `=180+60, 360-60`  
  `=240^@, 300^@`  

 
`theta-60^@=240^@\ \ =>\ \ theta=300^@`

`theta-60^@=300^@\ \ =>\ \ theta=360^@`

 
`text{Consider}\ theta = 0^@`

`sin(0-60^@)=sin(-60^@)=-sqrt3/2`

`:.theta=0^@, 300^@ and 360^@`

Advanced Trigonometry, 2ADV T2 SM-Bank 38

Solve  \(2\cos(2\theta) = -\sqrt{3}\)  for  \(\theta\), where  \(0^{\circ} \leq \theta \leq 180^{\circ}\).   (2 marks) 

Show Answers Only

\(\theta=75^{\circ}, 105^{\circ}\)

Show Worked Solution
\(2\cos(2\theta)\) \(= -\sqrt{3}\)  
\(\cos(2\theta)\) \(=-\dfrac{\sqrt{3}}{2}\)  

 
\(\text{Reference angle:}\ \cos\,30^{\circ}=\dfrac{\sqrt{3}}{2}\)

\(\text{Since cos is negative in 2nd/3rd quadrants:}\)

\(2\theta\) \(= 180-30, 180+30, 360+180-30, 360+180+30,\ …\)
  \(=150^{\circ}, 210^{\circ}, 510^{\circ},\ …\) 
\(\therefore \theta\) \(=75^{\circ}, 105^{\circ}\ \ \ (0^{\circ} \leq \theta \leq 180^{\circ})\)

Advanced Trigonometry, 2ADV T2 SM-Bank 37

Solve the equation  \(\cos\left(\dfrac{3\theta}{2}\right) = \dfrac{1}{2}\)  for \(-90^{\circ} \leq \theta \leq 90^{\circ}\).   (2 marks)

Show Answers Only

\(\theta = -40^{\circ}, 40^{\circ}\)

Show Worked Solution

\(\cos\left(\dfrac{3\theta}{2}\right) = \dfrac{1}{2}\)

\(\text{Reference angle:}\ \cos\,60^{\circ} = \dfrac{1}{2}\)

\(\text{Since cos is positive in 1st/4th quadrants:}\)

\(\dfrac{3\theta}{2}\) \(= -60^{\circ}, 60^{\circ}, 360^{\circ},\ …\)
\(\theta\) \(= -40^{\circ}, 40^{\circ}, 240^{\circ},\ …\)
  \(= -40^{\circ}, 40^{\circ}\ \ \ (-90^{\circ} \leq \theta\leq 90^{\circ})\)

Advanced Trigonometry, SMB-026

  1. Use the graph to estimate the two values of \(\theta\) where  \(\cos\,\theta=0.3\).   (1 mark)

  2. \(\cos 300^{\circ} \gt \sin 50^{\circ}\)
  3. Determine whether this statement is correct, using the graph to provide evidence for your answer.   (2 marks)

Show Answers Only

a. 

\(\text{Draw a horizontal line on graph from}\ \ \cos\,\theta=0.3:\)
 

\(\theta \approx 75^{\circ}, 285^{\circ}\)
 

b.   \(\cos\,300^{\circ}\ \text{is in the 4th quadrant.}\)

\(\text{Reference angle:}\ \theta=360-300=60^{\circ}\)

\(\Rightarrow \ \cos\,300^{\circ} = \cos\,60^{\circ} \)

\(\sin\,50^{\circ} = \cos(90-50)^{\circ} = \cos\,40^{\circ}\)
 

\(\text{Graphically, it can be seen that}\ \ \cos 60^{\circ} \lt \cos\,40^{\circ}\)

\(\Rightarrow\ \cos 300^{\circ} \lt \sin\,50^{\circ}\)

\(\therefore \ \text{Statement is not correct.}\)

Show Worked Solution

a.    \(\text{Draw a horizontal line on graph from}\ \ \cos\,\theta=0.3:\)
 

\(\theta \approx 75^{\circ}, 285^{\circ}\)
 

b.   \(\cos\,300^{\circ}\ \text{is in the 4th quadrant.}\)

\(\text{Reference angle:}\ \theta=360-300=60^{\circ}\)

\(\Rightarrow \ \cos\,300^{\circ} = \cos\,60^{\circ} \)

\(\sin\,50^{\circ} = \cos(90-50)^{\circ} = \cos\,40^{\circ}\)
 

\(\text{Graphically, it can be seen that}\ \ \cos 60^{\circ} \lt \cos\,40^{\circ}\)

\(\Rightarrow\ \cos 300^{\circ} \lt \sin\,50^{\circ}\)

\(\therefore \ \text{Statement is not correct.}\)

Advanced Trigonometry, SMB-025

 

  1. Use the graph to estimate the two values of \(\theta\) where  \(\sin\,\theta=0.4\).   (1 mark)
  2. Determine the largest value of \(\theta\) in the range  \(0^{\circ} \lt \theta \lt 360^{\circ}\), where  \(\sin\,\theta=-0.6\).   (1 marks)

Show Answers Only

a.   \(\text{Drawing a horizontal line from 0.4 on the \(y\)-axis:}\)
 

\(\therefore \theta \approx 25^{\circ}, 155^{\circ}\)
 

b.   \(\text{Drawing a horizontal line from –0.6 on the \(y\)-axis:}\)
 

\(\therefore \theta \approx 325^{\circ}\)

Show Worked Solution

a.   \(\text{Drawing a horizontal line from 0.4 on the \(y\)-axis:}\)
 

\(\therefore \theta \approx 25^{\circ}, 155^{\circ}\)
 

b.   \(\text{Drawing a horizontal line from –0.6 on the \(y\)-axis:}\)
 

\(\therefore \theta \approx 325^{\circ}\)

Advanced Trigonometry, SMB-024

  1. Use the graph to estimate the two values of \(\theta\) where  \(\cos\,\theta=-0.6\).   (1 mark)

  2. \(\cos 60^{\circ} \gt \sin 155^{\circ}\)
  3. Determine whether this statement is correct, using the graph to provide evidence for your answer.   (2 marks)

Show Answers Only

a.  \(\theta \approx 125^{\circ}, 235^{\circ}\)
 

       

b.   \(\sin\,155^{\circ}\ \text{is in the 2nd quadrant.}\)

\(\text{Reference angle:}\ \theta=180-155=25^{\circ}\)

\(\sin\,25^{\circ} =\cos(90-25)^{\circ}=\cos\,65^{\circ}\)
 

\(\text{Graphically, it is shown}\ \ \cos 60^{\circ} \lt \sin 155^{\circ} (\cos\,65^{\circ})\)

\(\therefore \ \text{Statement is not correct.}\)

Show Worked Solution

a.    \(\theta \approx 125^{\circ}, 235^{\circ}\)
 

b.   \(\sin\,155^{\circ}\ \text{is in the 2nd quadrant.}\)

\(\text{Reference angle:}\ \theta=180-155=25^{\circ}\)

\(\sin\,25^{\circ} =\cos(90-25)^{\circ}=\cos\,65^{\circ}\)

\(\text{Graphically, it is shown}\ \ \cos 60^{\circ} \gt \sin 155^{\circ} (\cos\,65^{\circ})\)

\(\therefore \ \text{Statement is correct.}\)

Advanced Trigonometry, SMB-023

  1. Use the graph to estimate the value of \(\sin\,\theta\) where  \(\theta=50^{\circ}\).   (1 mark)

  2. \(\cos 20^{\circ} \gt \sin 50^{\circ}\)
  3. Determine whether this statement is correct, using the graph to provide evidence for your answer.   (2 marks)

Show Answers Only

a.    \(\sin\,50^{\circ} \approx 0.8\)

b.   \(\cos\,20^{\circ}=\sin(90-20)^{\circ}=\sin\,70^{\circ}\)
 

\(\text{Graphically, it can be seen that}\ \ \cos 20^{\circ}\ (\sin\,70^{\circ}) \gt \sin 50^{\circ}\)

\(\therefore \ \text{Statement is correct.}\)

Show Worked Solution

a.    \(\sin\,50^{\circ} \approx 0.8\)

b.   \(\cos\,20^{\circ}=\sin(90-20)^{\circ}=\sin\,70^{\circ}\)
 

\(\text{Graphically, it can be seen that}\ \ \cos 20^{\circ}\ (\sin\,70^{\circ}) \gt \sin 50^{\circ}\)

\(\therefore \ \text{Statement is correct.}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 36

Solve the equation  \(\sin\left(\dfrac{x}{2}\right) = -\dfrac{1}{2}\)  for  \(360^{\circ} \leq x \leq 720^{\circ}\)   (2 marks)

Show Answers Only

\(x =420^{\circ}, 660^{\circ}\)

Show Worked Solution

\(\sin\left(\dfrac{x}{2}\right) = -\dfrac{1}{2}\)

\(\text{Reference angle:}\ \sin\,30^{\circ}=\dfrac{1}{2}\)

\(\text{Since sin is negative in 3rd/4th quarter:}\)

\(\dfrac{x}{2}\) \(=180+30, 360-30, 360+180+30,\ …\)
  \(=210^{\circ}, 330^{\circ}, 390^{\circ},\ …\)
\(x\) \(=420^{\circ}, 660^{\circ}, 780^{\circ},\ …\)

 
\(\text{Given}\ \ 360^{\circ} \leq x \leq 720^{\circ}:\)

\(\therefore x =420^{\circ}, 660^{\circ}\)

Advanced Trigonometry, 2ADV T2 2016 HSC 11g

Solve  \(\sin\left(\dfrac{x}{2}\right)= \dfrac{1}{2}\)  for  \(0^{\circ} \leq x \leq 360^{\circ}\).   (2 marks)

Show Answers Only

\(x = 60^{\circ}, 300^{\circ}\)

Show Worked Solution

\(\sin\left(\dfrac{x}{2}\right)= \dfrac{1}{2}\ \ \text{for}\ \ 0^{\circ} \leq x \leq 360^{\circ}\)

\(\Rightarrow\ \text{Reference angle}\ = 30^{\circ}\)

\(\dfrac{x}{2}\) \(= 30^{\circ}, 180-30, 360+30, …\)
  \(=30^{\circ}, 150^{\circ}, 390^{\circ},\ …\)

 

\(\therefore x = 60^{\circ}, 300^{\circ}\ \ \text{for}\ \ 0^{\circ} \leq x \leq 360^{\circ}\)

Advanced Trigonometry, 2ADV T2 2011 HSC 2b

Find the exact values of \(x\) such that  \(2\sin\,x =-\sqrt{3}\), where  \(0^{\circ} \leq x \leq 360^{\circ}\).   (2 marks)

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\(240^{\circ}, 300^{\circ}\)

Show Worked Solution
MARKER’S COMMENT: Better responses found the reference angle and then identified the correct quadrants, as shown.
\(2\sin\,x\) \(=-\sqrt{3}\ \ \text{where}\ \ 0^{\circ} \leq x \leq 360^{\circ}\)
\(\sin\,x\) \(= -\dfrac{\sqrt{3}}{2}\)
\(\sin\,60^{\circ}\) \(= \dfrac{\sqrt{3}}{2}\)

 

\(\text{Since}\ \sin\,x\ \text{is negative in 3rd/4th quadrants:}\)

\(x\) \(= 180+60,\ 360-60\)
  \(= 240^{\circ}, 300^{\circ}\)

Advanced Trigonometry, SMB-022

Complete the table of exact values.    (3 marks)

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ \ 0^{\circ} \ \ & 30^{\circ} & 45^{\circ} & 60^{\circ} & \ 90^{\circ} \  \\
\hline
\rule{0pt}{3.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& \quad & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \quad & 1 \\
\hline
\rule{0pt}{4ex} \cos \theta \rule[-2.5ex]{0pt}{0pt}& 1 & \dfrac{\sqrt{3}}{2} & \quad & \quad & 0 \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& 0 & \quad & \quad & \dfrac{1}{\sqrt{3}} & \infty \\
\hline
\end{array}

Show Answers Only

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ \ 0^{\circ} \ \ & 30^{\circ} & 45^{\circ} & 60^{\circ} & \ 90^{\circ} \  \\
\hline
\rule{0pt}{3.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}}  & \dfrac{\sqrt{3}}{2} & 1 \\
\hline
\rule{0pt}{4ex} \cos \theta \rule[-2.5ex]{0pt}{0pt}& 1 & \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} & 0 \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& 0 & \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} & \infty \\
\hline
\end{array}

Show Worked Solution

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ \ 0^{\circ} \ \ & 30^{\circ} & 45^{\circ} & 60^{\circ} & \ 90^{\circ} \  \\
\hline
\rule{0pt}{3.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}}  & \dfrac{\sqrt{3}}{2} & 1 \\
\hline
\rule{0pt}{4ex} \cos \theta \rule[-2.5ex]{0pt}{0pt}& 1 & \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} & 0 \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& 0 & \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} & \infty \\
\hline
\end{array}

Advanced Trigonometry, SMB-021

Complete the table of exact values.   (3 marks)

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ \ 0^{\circ} \ \ & 30^{\circ} & 45^{\circ} & 60^{\circ} & \ 90^{\circ} \  \\
\hline
\rule{0pt}{3.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& 0 & \quad & \quad  & \dfrac{\sqrt{3}}{2} & 1 \\
\hline
\rule{0pt}{4ex} \cos \theta \rule[-2.5ex]{0pt}{0pt}& \quad & \quad & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} & 0 \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& \quad & \dfrac{1}{\sqrt{3}} & 1 & \quad & \infty \\
\hline
\end{array}

Show Answers Only

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ \ 0^{\circ} \ \ & 30^{\circ} & 45^{\circ} & 60^{\circ} & \ 90^{\circ} \  \\
\hline
\rule{0pt}{3.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}}  & \dfrac{\sqrt{3}}{2} & 1 \\
\hline
\rule{0pt}{4ex} \cos \theta \rule[-2.5ex]{0pt}{0pt}& 1 & \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} & 0 \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& 0 & \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} & \infty \\
\hline
\end{array}

Show Worked Solution

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ \ 0^{\circ} \ \ & 30^{\circ} & 45^{\circ} & 60^{\circ} & \ 90^{\circ} \  \\
\hline
\rule{0pt}{3.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}}  & \dfrac{\sqrt{3}}{2} & 1 \\
\hline
\rule{0pt}{4ex} \cos \theta \rule[-2.5ex]{0pt}{0pt}& 1 & \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} & 0 \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& 0 & \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} & \infty \\
\hline
\end{array}

Advanced Trigonometry, SMB-020

Using the right-angled triangles above, or otherwise, complete the table below.   (3 marks)

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ 30^{\circ} \ & \ 45^{\circ} \ & \ 60^{\circ}\ \\
\hline
\rule{0pt}{4.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} &\quad\\
\hline
\rule{0pt}{4.5ex} \cos \theta \rule[-3ex]{0pt}{0pt}& \dfrac{\sqrt{3}}{2}& \quad & \quad \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-2.5ex]{0pt}{0pt}& \quad & 1 & \sqrt{3} \\
\hline
\end{array}

Show Answers Only

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ 30^{\circ} \ & \ 45^{\circ} \ & \ 60^{\circ}\ \\
\hline
\rule{0pt}{4.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} \\
\hline
\rule{0pt}{4.5ex} \cos \theta \rule[-3ex]{0pt}{0pt}& \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} \\
\hline
\end{array}

Show Worked Solution

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ 30^{\circ} \ & \ 45^{\circ} \ & \ 60^{\circ}\ \\
\hline
\rule{0pt}{4.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} \\
\hline
\rule{0pt}{4.5ex} \cos \theta \rule[-3ex]{0pt}{0pt}& \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} \\
\hline
\end{array}

Advanced Trigonometry, SMB-019

Using the right-angled triangles above, or otherwise, complete the table below.   (3 marks)

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ 30^{\circ} \ & \ 45^{\circ} \ & \ 60^{\circ}\ \\
\hline
\rule{0pt}{4.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& \quad & \quad &\quad\\
\hline
\rule{0pt}{4.5ex} \cos \theta \rule[-3ex]{0pt}{0pt}& \quad & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{\sqrt{3}} & 1 & \quad \\
\hline
\end{array}

Show Answers Only

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ 30^{\circ} \ & \ 45^{\circ} \ & \ 60^{\circ}\ \\
\hline
\rule{0pt}{4.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} \\
\hline
\rule{0pt}{4.5ex} \cos \theta \rule[-3ex]{0pt}{0pt}& \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} \\
\hline
\end{array}

Show Worked Solution

\begin{array}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \theta \rule[-1ex]{0pt}{0pt}& \ 30^{\circ} \ & \ 45^{\circ} \ & \ 60^{\circ}\ \\
\hline
\rule{0pt}{4.5ex} \sin \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} \\
\hline
\rule{0pt}{4.5ex} \cos \theta \rule[-3ex]{0pt}{0pt}& \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} \\
\hline
\rule{0pt}{3.5ex} \tan \theta \rule[-3ex]{0pt}{0pt}& \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} \\
\hline
\end{array}

Advanced Trigonometry, 2ADV T2 SM-Bank 43v4

Find the exact value of

\(\sin(-300^{\circ})\).   (2 marks)

Show Answers Only

\(\dfrac{\sqrt{3}}{2}\)

Show Worked Solution

\(\sin(-300^{\circ})= \sin\,60^{\circ}\)

\(\text{Reference angle:}\ 60^{\circ}\)

\(\text{Since sin is positive in 1st quadrant:}\)

\(\sin(-300^{\circ})= \sin\,60^{\circ}=\dfrac{\sqrt{3}}{2}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 43v3

Find the exact value of

\(\sin(-210^{\circ})\).   (2 marks)

Show Answers Only

\(\dfrac{1}{2}\)

Show Worked Solution

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

\(\text{Reference angle:}\ 180-150=30^{\circ}\)

\(\text{Since sin is positive in 2nd quadrant:}\)

\(\sin(-210^{\circ})= \sin\,30^{\circ}=\dfrac{1}{2}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 43v2

Find the exact value of

\(\cos(-240^{\circ})\).   (2 marks)

Show Answers Only

\(-\dfrac{1}{2}\)

Show Worked Solution

\(\cos(-240^{\circ})= \cos\,120^{\circ}\)

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

\(\text{Since cos is negative in 2nd quadrant:}\)

\(\cos(-240^{\circ})= -\cos\,60^{\circ}=-\dfrac{1}{2}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 43v1

Find the exact value of

\(\tan(-150^{\circ})\).   (2 marks)

Show Answers Only

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

Show Worked Solution

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

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

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

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

Advanced Trigonometry, 2ADV T2 2015 HSC 12a

Find the solutions of  \(2\sin\,\theta = 1\)  for  \(0^{\circ} \leq \theta \leq 360^{\circ}\).   (2 marks)

Show Answers Only

\(30^{\circ}, 150^{\circ}\)

Show Worked Solution

\(2\sin\,\theta=1\ \ \Rightarrow\ \ \sin\,\theta=\dfrac{1}{2}\)

\(\text{Reference angle:}\ \sin\,30^{\circ}=\dfrac{1}{2}\)

\(\text{Since sin is positive in the 1st/2nd quadrants:}\)

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

Advanced Trigonometry, 2ADV T2 2009 HSC 1e

Find the exact value of \(\theta\) such that  \(2\cos\,\theta = 1\), where  \(0^{\circ} \leq \theta \leq 90^{\circ}\).   (2 marks)

Show Answers Only

 \(\theta = 60^{\circ}\)

Show Worked Solution
\(2 \cos\,\theta\) \(= 1\)
\(\cos\,\theta\) \(= \dfrac{1}{2}\)
\(\therefore \theta\) \(= 60^{\circ},\ \ \ \ 0^{\circ} \leq \theta \leq 90^{\circ}\)

Advanced Trigonometry, 2ADV T2 2007 HSC 4a

Solve  \(\sqrt{2}\,\sin\,x = 1\)  for  \(0^{\circ} \leq x \leq 360^{\circ}.\)   (2 marks)

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\(45^{\circ}, 135^{\circ}\)

Show Worked Solution

\(\sqrt{2}\,\sin\,x = 1\ \ \Rightarrow \ \ \sin\,x= \dfrac{1}{\sqrt{2}} \)

\(\text{Reference angle:}\ \sin\,45^{\circ} = \dfrac{1}{\sqrt{2}}\)

\(\text{Since sin is positive in 1st/2nd quadrants:}\)

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

Advanced Trigonometry, 2ADV T2 2005 HSC 2a

Solve  \(\cos\,\theta = \dfrac{1}{\sqrt{2}}\)  for  \(0^{\circ} ≤ \theta ≤ 360^{\circ}\).   (2 marks)

Show Answers Only

\(45^{\circ}, 315^{\circ}`

Show Worked Solution

\(\cos\,\theta = \dfrac{1}{\sqrt{2}}\)  for  \(0^{\circ} ≤ \theta ≤ 360^{\circ}\)

\(\text{Reference angle:}\ \cos\,45^{\circ} = \dfrac{1}{\sqrt{2}}\)

\(\text{Since cos is positive in 1st/4th quadrants:}\)

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

 

Advanced 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. \(6\)
Show Answers Only

\(C\)

Show Worked Solution

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

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

 
\(\Rightarrow C\)

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^@` 
Show Answers Only

`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, 2ADV T1 2013 HSC 2v3 MC

The diagram shows the line \(\ell\).
 

 What is the slope of the line \(\ell\) ? 

  1. \(\dfrac{1}{\sqrt2}\)
  2. \(-\dfrac{1}{\sqrt2}\)
  3. \(1\)
  4. \(-1\)
Show Answers Only

\(C\)

Show Worked Solution

\(\text{Gradient is positive (slopes from bottom left to top right).}\)

\(\tan\,45^{\circ} = 1\)

\(\therefore\ \text{Gradient is 1}\)

\(\Rightarrow C\)

Advanced Trigonometry, 2ADV T1 2013 HSC 2v2 MC

The diagram shows the line \(\ell\).
 

 What is the slope of the line \(\ell\) ? 

  1. \(\sqrt3\)
  2. \(-\sqrt3\)
  3. \(\dfrac{1}{\sqrt3}\)
  4. \(-\dfrac{1}{\sqrt3}\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Gradient is negative (slopes from top left to bottom right).}\)

\(\text{The line cuts the \(x\)-axis at an acute angle =}\ 90-60=30^{\circ}\) 

\(\tan\,30^{\circ} = \dfrac{1}{\sqrt3}\)

\(\therefore\ \text{Gradient is}\ -\dfrac{1}{\sqrt3}\)

\(\Rightarrow D\)

Advanced Trigonometry, SMB-012 MC

In which quadrant does \(\theta\) lie, given the following information:

\(\tan \theta \lt 0,\ \ \cos \theta \lt 0\)

  1. 1st Quadrant
  2. 2nd Quadrant
  3. 3rd Quadrant
  4. 4th Quadrant
Show Answers Only

\(B\)

Show Worked Solution

\(\tan \theta \lt 0\ \ \Rightarrow\ \ \text{2nd/4th quadrants}\)

\(\cos \theta \lt 0\ \ \Rightarrow\ \ \text{2nd/3rd quadrants}\)

\(\theta\ \text{must be in the 2nd quadrant}\)

\(\Rightarrow B\)

Advanced Trigonometry, SMB-011 MC

In which quadrant does \(\theta\) lie, given the following information:

\(\sin \theta \lt 0,\ \ \cos \theta \gt 0\)

  1. 1st Quadrant
  2. 2nd Quadrant
  3. 3rd Quadrant
  4. 4th Quadrant
Show Answers Only

\(D\)

Show Worked Solution

\(\sin \theta \lt 0\ \ \Rightarrow\ \ \text{3rd/4th quadrants}\)

\(\cos \theta \gt 0\ \ \Rightarrow\ \ \text{1st/4th quadrants}\)

\(\theta\ \text{must be in the 4th quadrant}\)

\(\Rightarrow D\)

Advanced Trigonometry, SMB-018

Express each of the following in terms of its reference angle \(\theta\), where  \(0^{\circ} \leq \theta \leq 90^{\circ}\).

  1. \(\tan 175^{\circ}\).   (1 mark)

  2. \(\cos 262^{\circ}\).   (1 mark)

Show Answers Only

a.   \(-\tan 5^{\circ}\)

b.   \(-\cos 82^{\circ}\)

Show Worked Solution

a.   \(175^{\circ}\ \ \Rightarrow\ \ \text{2nd quadrant (sin +ve/cos –ve)} \)

\(\text{Reference angle:}\ 180-175=5^{\circ}\)

\(\tan 175^{\circ} = \dfrac{\sin 5^{\circ}}{-\cos 5^{\circ}} = -\tan 5^{\circ}\)
 

b.   \(262^{\circ}\ \ \Rightarrow\ \ \text{3rd quadrant (cos –ve)} \)

\(\text{Reference angle:}\ \ 180+\theta=262^{\circ}\ \ \Rightarrow\ \ \theta=82^{\circ}\)

\(\cos 262^{\circ} = -\cos 82^{\circ}\)

Advanced Trigonometry, SMB-017

Express each of the following in terms of its reference angle \(\theta\), where  \(0^{\circ} \leq \theta \leq 90^{\circ}\).

  1. \(\cos 140^{\circ}\).   (1 mark)

  2. \(\tan 345^{\circ}\).   (1 mark)

Show Answers Only

a.   \(-\cos 40^{\circ}\)

b.  \(-\tan 15^{\circ}\)

Show Worked Solution

a.   \(140^{\circ}\ \ \Rightarrow\ \ \text{2nd quadrant (cos –ve)} \)

\(\text{Reference angle:}\ \ 180-140=40^{\circ}\)

\(\cos 140^{\circ} = -\cos 40^{\circ}\)

 

b.   \(345^{\circ}\ \ \Rightarrow\ \ \text{4th quadrant (sin –ve/cos +ve)} \)

\(\text{Reference angle:}\ 360-345=15^{\circ}\ \ \Rightarrow \ \theta=15^{\circ}\)

\(\tan 345^{\circ} = \dfrac{-\sin 15^{\circ}}{\cos 15^{\circ}} = -\tan 15^{\circ}\)