If `sin theta =-(1)/sqrt(2)` and `tan theta=1` then `theta` lies in which quadrant

Options

A. first
B. second
C. third
D. fourth

Answer

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

### Step 1: Analyze the given values
We know:
- \( \sin \theta = -\frac{1}{\sqrt{2}} \)
- \( \tan \theta = 1 \)

### Step 2: Identify the sign of sine and tangent
From the values:
- Since \( \sin \theta \) is negative, \( \theta \) must be in either the third or fourth quadrant (as sine is negative in these quadrants).
- Since \( \tan \theta \) is positive, \( \theta \) must be in the third quadrant (as tangent is positive in the third quadrant).

### Step 3: Conclusion
Given that \( \sin \theta \) is negative and \( \tan \theta \) is positive, we conclude that \( \theta \) lies in the **third quadrant**.

### Final Answer
Thus, the answer is that \( \theta \) lies in the **third quadrant**.

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