In `Delta PQR, Q` is a right angle, PQ = 3 and QR = 4. If `angle P = alpha and R = beta`, then tan `beta` is equal to

Options

A. `(4)/(3)`
B. `(3)/(4)`
C. `(3)/(5)`
D. `(5)/(3)`

Answer

To solve the problem, we need to determine the value of \(\tan \beta\) in triangle \(PQR\), where \(Q\) is the right angle, \(PQ = 3\), and \(QR = 4\).

### Step-by-Step Solution:

1. **Identify the Triangle and its Sides**:
- We have triangle \(PQR\) with \(Q\) as the right angle.
- The sides are given as \(PQ = 3\) (the side opposite angle \(R\)) and \(QR = 4\) (the side opposite angle \(P\)).

2. **Determine the Relationship for \(\tan \beta\)**:
- In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
- For angle \(R\) (which is \(\beta\)), the opposite side is \(PQ\) and the adjacent side is \(QR\).

3. **Write the Formula for \(\tan \beta\)**:
\[ \tan \beta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{PQ}{QR} \]

4. **Substitute the Values**:
- We know \(PQ = 3\) and \(QR = 4\).
\[ \tan \beta = \frac{3}{4} \]

5. **Final Result**:
- Therefore, \(\tan \beta = \frac{3}{4}\).