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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
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\(B\)

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

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

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

\(\Rightarrow B\)

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
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\(D\)

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-014

Determine the quadrant in which the following statement about \(\theta\) is true:

  1. \(\cos \theta\) is negative and \(\sin \theta\) is negative.   (1 mark)

  2. \(\tan \theta\) is negative and \(\cos \theta\) is positive.   (1 mark)

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a.   \(\text{3rd quadrant}\)

b.   \(\text{4th quadrant}\)

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a.   \(\cos \theta \ \Rightarrow\ \text{–ve in 2nd/3rd quadrant}\)

\(\sin \theta \ \Rightarrow\ \text{–ve in 3rd/4th quadrant}\)

\(\Rightarrow\ \text{3rd quadrant}\)
  

b.   \(\tan \theta \ \Rightarrow\ \text{–ve in 2nd/4th quadrant}\)

\(\cos \theta \ \Rightarrow\ \text{+ve in 1st/4th quadrant}\)

\(\Rightarrow\ \text{4th quadrant}\)

Advanced Trigonometry, SMB-013

Determine the quadrant in which the following statement about \(\theta\) is true:

  1. \(\sin \theta\) is positive and \(\cos \theta\) is positive.   (1 mark)

  2. \(\tan \theta\) is negative and \(\cos \theta\) is negative.   (1 mark)

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a.   \(\text{1st quadrant}\)

b.   \(\text{2nd quadrant}\)

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a.   \(\sin \theta \ \Rightarrow\ \text{+ve in 1st/2nd quadrant}\)

\(\cos \theta \ \Rightarrow\ \text{+ve in 1st/4th quadrant}\)

\(\Rightarrow\ \text{1st quadrant}\)
  

b.   \(\tan \theta \ \Rightarrow\ \text{–ve in 2nd/4th quadrant}\)

\(\cos \theta \ \Rightarrow\ \text{–ve in 2nd/3rd quadrant}\)

\(\Rightarrow\ \text{2nd quadrant}\)

Advanced Trigonometry, SMB-012

Determine the quadrant in which the following statement about \(\theta\) is true:

  1. \(\tan \theta\) is positive and \(\sin \theta\) is negative.   (1 mark)

  2. \(\sin \theta\) is negative and \(\cos \theta\) is positive.   (1 mark)

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a.   \(\text{3rd quadrant}\)

b.   \(\text{4th quadrant}\)

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a.  \(\tan \theta \ \Rightarrow\ \text{+ve in 1st/3rd quadrant}\)

\(\sin \theta \ \Rightarrow\ \text{–ve in 3rd/4th quadrant}\)

\(\Rightarrow\ \theta\ \text{is in the 3rd quadrant}\)
  

b.   \(\sin \theta \ \Rightarrow\ \text{–ve in 3rd/4th quadrant}\)

\(\cos \theta \ \Rightarrow\ \text{+ve in 1st/4th quadrant}\)

\(\Rightarrow\ \theta\ \text{is in the 4th quadrant}\)

Advanced Trigonometry, SMB-008 MC

Consider the angle  \(\theta = 290^{\circ}\).

Which of the following correctly describes the quadrant containing this angle?

  1. The quadrant where \(\cos \theta\) is negative and \(\sin \theta\) is positive.
  2. The quadrant where \(\cos \theta\) is negative and \(\sin \theta\) is negative.
  3. The quadrant where \(\tan \theta\) is negative and \(\cos \theta\) is positive.
  4. The quadrant where \(\cos \theta\) is positive and \(\sin \theta\) is negative.
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\(D\)

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\(\text{The angle 290° is in the 4th quadrant}\ (270^{\circ} \lt 290^{\circ} \lt 360^{\circ})\)

\(\text{Quadrant IV:}\ \sin(-), \cos (+), \tan(-) \)

\(\Rightarrow D\)

Advanced Trigonometry, SMB-007 MC

Consider the angle \(\theta = 135^{\circ}\).

Which of the following correctly describes the quadrant in which this angle lies?

  1. The quadrant where \(\cos \theta\) is negative and \(\sin \theta\) is positive.
  2. The quadrant where \(\cos \theta\) is positive and \(\sin \theta\) is negative.
  3. The quadrant where \(\tan \theta\) is positive and \(\sin \theta\) is negative.
  4. The quadrant where \(\cos \theta\) is negative and \(\sin \theta\) is negative.
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\(A\)

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\(\text{The angle 135° is in the 2nd quadrant.}\)

\(\text{Quadrant II:}\  \sin (+), \cos(-), \tan(-) \)

\(\Rightarrow A\)

Advanced Trigonometry, SMB-006 MC

Consider the angle  \(\theta = 260^{\circ}\)

Determine which of the following descriptions correctly identify the quadrant containing this angle.

  1. The quadrant where \(\cos \theta\) is negative and \(\sin \theta\) is positive.
  2. The quadrant where \(\cos \theta\) is positive and \(\tan \theta\) is positive.
  3. The quadrant where \(\tan \theta\) is negative and \(\sin \theta\) is positive.
  4. The quadrant where \(\cos \theta\) is negative and \(\sin \theta\) is negative.
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\(D\)

Show Worked Solution

\(260^{\circ}\ \text{lies in Quadrant III}\ \ (180^{\circ} \lt 260^{\circ} \lt 270^{\circ}) \)

\(\text{In quadrant III:}\ \ \tan (+), \sin (-), \cos (-)\)

\(\Rightarrow D\)

Advanced Trigonometry, SMB-007

A point \(X\) is located on the unit circle at angle 300\(^{\circ}\) from the positive \(x\)-axis.

  1. Determine the quadrant in which \(X\) lies.   (1 mark)

  2. Find the coordinates of point \(X\).   (2 marks)

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a.   \(\text{Quadrant IV}\)

b.   \(X\left( \dfrac{1}{2}, -\dfrac{\sqrt{3}}{2}\right) \)

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a.   
         

\(300^{\circ}\ \text{lies in Quadrant IV}\ \ (270^{\circ} \lt 300^{\circ} \lt 360^{\circ}) \)
 

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

\(x\text{-coordinate of}\ X = \cos\,60^{\circ} = \dfrac{1}{2}\)

\(y\text{-coordinate of}\ X = -\sin\,60^{\circ} = -\dfrac{\sqrt{3}}{2}\)

\(X\left( \dfrac{1}{2}, -\dfrac{\sqrt{3}}{2}\right) \)

Advanced Trigonometry, SMB-006

A point \(V\) is located on the unit circle at angle 240\(^{\circ}\) from the positive \(x\)-axis.

  1. Determine the quadrant in which \(V\) lies.   (1 mark)

  2. Find the coordinates of point \(V\).   (2 marks)

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a.   \(\text{Quadrant III}\)

b.   \(V\left(-\dfrac{1}{2}, -\dfrac{\sqrt{3}}{2}\right) \)

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a.   
         

\(240^{\circ}\ \text{lies in Quadrant III}\ \ (180^{\circ} \lt 240^{\circ} \lt 270^{\circ}) \)
 

b.   \(\text{Reference angle}\ (\theta):\  180^{\circ}+\theta=240^{\circ}\ \ \Rightarrow \ \ \theta=60^{\circ}\)

\(x\text{-coordinate of}\ V = -\cos\,60^{\circ} = -\dfrac{1}{2}\)

\(y\text{-coordinate of}\ V = -\sin\,60^{\circ} = -\dfrac{\sqrt{3}}{2}\)

\(V\left(-\dfrac{1}{2}, -\dfrac{\sqrt{3}}{2}\right) \)

Advanced Trigonometry, SMB-001

A point \(Q\) is located on the unit circle at angle 150\(^{\circ}\) from the positive \(x\)-axis.

  1. Determine the quadrant in which \(Q\) lies.   (1 mark)

  2. Find the coordinates of point \(Q\).   (2 marks)

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a.   \(\text{Quadrant II}\)

b.   \(Q\left( -\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) \)

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a.   \(150^{\circ}\ \text{lies in Quadrant II}\ \ \ (90^{\circ} \lt 150^{\circ} \lt 180^{\circ})\)
 

b.  
         

\(x\text{-coordinate}\ = -\cos\,30^{\circ} = -\dfrac{\sqrt{3}}{2}\)

\(y\text{-coordinate}\ = \sin\,30^{\circ} = \dfrac{1}{2}\)

\(Q\left( -\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) \)

Advanced Trigonometry, SMB-002 MC

If point \(P\) lies on the unit circle at coordinates \((0.5, -0.866)\), which quadrant does \(P\) lie in and what is the approximate angle in standard position?

  1. Quadrant \(\text{III}\), 210\(^{\circ}\)
  2. Quadrant \(\text{IV}\), 300\(^{\circ}\)
  3. Quadrant \(\text{III}\), 240\(^{\circ}\)
  4. Quadrant \(\text{IV}\), 330\(^{\circ}\)
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\(B\)

Show Worked Solution

\(P\ \text{is found in Quadrant IV}\)
 

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

\(\text{Angle in standard position}\ =360-60 = 300^{\circ}\)

\(\Rightarrow B\)