If `sintheta=-(1)/(2)andtantheta=(1)/(sqrt(3))`, then in which quadrant does `theta` lie ?

Options

A. First
B. Second
C. Third
D. Fourth

Explanation

To determine in which quadrant the angle \( \theta \) lies given that \( \sin \theta = -\frac{1}{2} \) and \( \tan \theta = \frac{1}{\sqrt{3}} \), we can follow these steps:

### Step 1: Analyze the given values of sine and tangent
We are given:
- \( \sin \theta = -\frac{1}{2} \) (which is negative)
- \( \tan \theta = \frac{1}{\sqrt{3}} \) (which is positive)

### Step 2: Recall the signs of trigonometric functions in each quadrant
In the four quadrants:
- **Quadrant I**: All functions (sine, cosine, tangent) are positive.
- **Quadrant II**: Sine is positive, cosine and tangent are negative.
- **Quadrant III**: Sine is negative, cosine is negative, tangent is positive.
- **Quadrant IV**: Sine is negative, cosine is positive, tangent is negative.

### Step 3: Determine the quadrant based on the signs
From the values:
- Since \( \sin \theta \) is negative, \( \theta \) cannot be in Quadrant I or II.
- Since \( \tan \theta \) is positive, \( \theta \) cannot be in Quadrant IV.

Thus, the only quadrant where sine is negative and tangent is positive is Quadrant III.

### Conclusion
Therefore, \( \theta \) lies in the **Third Quadrant**.

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