If `tan theta=(4)/(3)`, in which quadrant does `theta` lie ?

Answer

To determine in which quadrant the angle \( \theta \) lies given that \( \tan \theta = \frac{4}{3} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Tangent Function**:
The tangent function, \( \tan \theta \), is defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be represented as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

2. **Identifying the Sign of Tangent**:
We are given that \( \tan \theta = \frac{4}{3} \), which is a positive value.

3. **Quadrants and the Sign of Tangent**:
The tangent function is positive in the following quadrants:
- **First Quadrant**: \( 0 < \theta < \frac{\pi}{2} \)
- **Third Quadrant**: \( \pi < \theta < \frac{3\pi}{2} \)

4. **Conclusion**:
Since \( \tan \theta \) is positive, \( \theta \) must lie in either the first quadrant or the third quadrant.

### Final Answer:
Thus, \( \theta \) lies in the **first quadrant** or the **third quadrant**.

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