Question

In which quadrant would \theta be if \tan \theta = -√(3) and \sin \theta \ \textgreater \ 0?
A. Quadrant IV
B. Quadrant I
C. Quadrant III
D. Quadrant II

Answer 1

To determine the quadrant where \(\theta\) would be given that \(\tan \theta = -\sqrt{3}\) and \(\sin \theta > 0\), we need to analyze the signs of the trigonometric functions based on the properties of each quadrant:

1. **Tangent (\(\tan \theta\))** is negative in Quadrants II and IV.
2. **Sine (\(\sin \theta\))** is positive in Quadrants I and II.

Since \(\tan \theta = -\sqrt{3}\) indicates that \(\theta\) can be in Quadrants II or IV, we must also take into account that \(\sin \theta > 0\), which narrows down our options.

- In **Quadrant II**, \(\tan \theta\) is negative and \(\sin \theta\) is positive.
- In **Quadrant IV**, \(\tan \theta\) is negative but \(\sin \theta\) is negative.

Given \(\sin \theta > 0\), \(\theta\) cannot be in Quadrant IV.

Thus, \(\theta\) must be in **Quadrant II**.

The answer is **D. Quadrant II**.