PQ is a part of a given height a and AB is a tower at some distance, `alpha and beta` are the angles of elevation of B, the top of the tower at P and Q respectively. The height of the tower is

Options

A. `(asinalpha sinbeta)/(sin(alpha-beta))`
B. `(acosalphacosbeta)/(sin(alpha-beta))`
C. `(asinalphacosbeta)/(sin(alpha-beta))`
D. None

Answer

To solve the problem of finding the height of the tower (AB) given the angles of elevation (α and β) from points P and Q respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Let the height of the part PQ be \( A \).
- Let the height of the tower AB be \( H \).
- The angles of elevation from point P to the top of the tower (B) is \( \alpha \).
- The angles of elevation from point Q to the top of the tower (B) is \( \beta \).

2. **Set Up the Right Triangles:**
- From point P, we can form a right triangle \( \triangle ABP \) where:
- \( \tan(\alpha) = \frac{H}{AP} \) (where \( AP \) is the horizontal distance from A to P).
- From point Q, we can form another right triangle \( \triangle BRQ \) where:
- \( \tan(\beta) = \frac{H - A}{QR} \) (where \( QR \) is the horizontal distance from B to Q).

3. **Express Distances in Terms of Angles:**
- From triangle \( \triangle ABP \):
\[ AP = \frac{H}{\tan(\alpha)} \]
- From triangle \( \triangle BRQ \):
\[ QR = \frac{H - A}{\tan(\beta)} \]

4. **Equate the Horizontal Distances:**
- Since \( AP = QR \), we can set the two expressions equal:
\[ \frac{H}{\tan(\alpha)} = \frac{H - A}{\tan(\beta)} \]

5. **Cross-Multiply and Rearrange:**
- Cross-multiplying gives:
\[ H \cdot \tan(\beta) = (H - A) \cdot \tan(\alpha) \]
- Expanding this results in:
\[ H \cdot \tan(\beta) = H \cdot \tan(\alpha) - A \cdot \tan(\alpha) \]

6. **Isolate H:**
- Rearranging the equation to isolate \( H \):
\[ H \cdot (\tan(\beta) - \tan(\alpha)) = -A \cdot \tan(\alpha) \]
- Thus,
\[ H = \frac{A \cdot \tan(\alpha)}{\tan(\alpha) - \tan(\beta)} \]

7. **Convert to Sine and Cosine:**
- Using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), we can rewrite:
\[ H = \frac{A \cdot \frac{\sin(\alpha)}{\cos(\alpha)}}{\frac{\sin(\alpha)}{\cos(\alpha)} - \frac{\sin(\beta)}{\cos(\beta)}} \]
- Simplifying gives:
\[ H = \frac{A \cdot \sin(\alpha) \cdot \cos(\beta)}{\sin(\alpha) \cdot \cos(\beta) - \sin(\beta) \cdot \cos(\alpha)} \]

8. **Final Result:**
- The height of the tower is:
\[ H = \frac{A \cdot \sin(\alpha) \cdot \cos(\beta)}{\sin(\alpha - \beta)} \]