If `sin theta = -1/2 and cos theta = (sqrt3)/(2),` then `theta ` lies in

Options

A. `1^(st)` quadrant
B. `II^(nd)` quadrant
C. `III^(rd)` quadrant
D. `IV^(th)` quadrant

Answer

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

### Step-by-Step Solution:

1. **Identify the signs of sine and cosine:**
- We know that \( \sin \theta = -\frac{1}{2} \) indicates that sine is negative.
- We also have \( \cos \theta = \frac{\sqrt{3}}{2} \), which indicates that cosine is positive.

2. **Determine the quadrants based on the signs:**
- In the **first quadrant**, both sine and cosine are positive.
- In the **second quadrant**, sine is positive and cosine is negative.
- In the **third quadrant**, both sine and cosine are negative.
- In the **fourth quadrant**, sine is negative and cosine is positive.

3. **Conclusion from the signs:**
- Since \( \sin \theta \) is negative and \( \cos \theta \) is positive, \( \theta \) must lie in the **fourth quadrant**.

4. **Verification using tangent:**
- We can also calculate \( \tan \theta \) to confirm:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{-1}{\sqrt{3}} \]
- Since \( \tan \theta \) is negative, this further confirms that \( \theta \) is in the fourth quadrant, where sine is negative and cosine is positive.

5. **Final answer:**
- Therefore, the angle \( \theta \) lies in the **fourth quadrant**.