Question

in which quadrant would \\( \theta \\) be if \\( \tan \theta = - \sqrt { 3 } \\) and \\( \sin \theta > 0 \\)? (1 point)

quadrant iv

quadrant iii

quadrant ii

quadrant i

Explanation:

To determine the quadrant of \(\theta\), we analyze the signs of \(\tan\theta\) and \(\sin\theta\):

1. Recall the sign rules for trigonometric functions in each quadrant:

• In Quadrant I: \(\sin\theta>0\), \(\cos\theta>0\), \(\tan\theta = \frac{\sin\theta}{\cos\theta}>0\).
• In Quadrant II: \(\sin\theta>0\), \(\cos\theta<0\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}<0\).
• In Quadrant III: \(\sin\theta<0\), \(\cos\theta<0\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}>0\).
• In Quadrant IV: \(\sin\theta<0\), \(\cos\theta>0\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}<0\).

1. Given \(\tan\theta = -\sqrt{3}<0\), so \(\theta\) could be in Quadrant II or IV (where \(\tan\theta\) is negative).
2. Given \(\sin\theta>0\), so \(\theta\) could be in Quadrant I or II (where \(\sin\theta\) is positive).
3. The intersection of the two conditions ( \(\tan\theta<0\) and \(\sin\theta>0\)) is Quadrant II.

Answer:

C. Quadrant II