Question

in which quadrant would \\( \theta \\) be if \\( \tan \theta = \frac { \sqrt { 3 } } { 3 } \\) and \\( \sin \theta = - \frac { 1 } { 2 } \\)? use a digit for your response rather than a roman numeral. (1 point)
quadrant
check answer remaining attempts : 3

Explanation:

Step1: Recall the signs of trigonometric functions in each quadrant

• In Quadrant I: \(\sin\theta> 0\), \(\cos\theta>0\), \(\tan\theta>0\)
• In Quadrant II: \(\sin\theta> 0\), \(\cos\theta<0\), \(\tan\theta<0\)
• In Quadrant III: \(\sin\theta< 0\), \(\cos\theta<0\), \(\tan\theta>0\)
• In Quadrant IV: \(\sin\theta< 0\), \(\cos\theta>0\), \(\tan\theta<0\)

Step2: Analyze the given trigonometric values

Given \(\tan\theta=\frac{\sqrt{3}}{3}>0\) and \(\sin\theta =-\frac{1}{2}<0\)
Since \(\tan\theta>0\), \(\theta\) is in either Quadrant I or Quadrant III.
Since \(\sin\theta<0\), \(\theta\) is in either Quadrant III or Quadrant IV.

Step3: Find the common quadrant

The common quadrant for \(\tan\theta>0\) and \(\sin\theta<0\) is Quadrant III.

Answer:

\(3\)