At a point on a level plane subtends an angle `theta` and flag staff of height a at the top of the tower subtends an angle `phi`. Find the height of the tower.

Answer

To find the height of the tower given the angles subtended by the tower and the flagstaff, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Problem**:
- We have a tower of height \( h \) and a flagstaff of height \( a \) on top of the tower.
- From a point \( P \) on the level plane, the angle subtended by the tower is \( \theta \) and the angle subtended by the top of the flagstaff is \( \phi \).

2. **Drawing the Diagram**:
- Draw a vertical line representing the tower (height \( h \)).
- Draw another vertical line on top of the tower representing the flagstaff (height \( a \)).
- Mark the point \( P \) from where the angles are measured.

3. **Identifying Triangles**:
- Let \( C \) be the top of the tower and \( B \) be the top of the flagstaff.
- The point \( A \) is at the base of the tower.
- We will analyze two triangles: \( \triangle CPB \) (for the tower) and \( \triangle APB \) (for the flagstaff).

4. **Using Tangent for Angles**:
- For triangle \( CPB \):
\[ \tan(\theta) = \frac{h}{d} \quad \text{(1)} \]
- For triangle \( APB \):
\[ \tan(\phi + \theta) = \frac{a + h}{d} \quad \text{(2)} \]

5. **Expressing \( d \)**:
- From equation (1), we can express \( d \):
\[ d = \frac{h}{\tan(\theta)} \quad \text{(3)} \]

6. **Substituting \( d \) in Equation (2)**:
- Substitute \( d \) from equation (3) into equation (2):
\[ \tan(\phi + \theta) = \frac{a + h}{\frac{h}{\tan(\theta)}} \]
- This simplifies to:
\[ \tan(\phi + \theta) = \frac{(a + h) \tan(\theta)}{h} \]

7. **Cross Multiplying**:
- Cross-multiply to eliminate the fraction:
\[ h \tan(\phi + \theta) = (a + h) \tan(\theta) \]

8. **Rearranging the Equation**:
- Rearranging gives:
\[ h \tan(\phi + \theta) - h \tan(\theta) = a \tan(\theta) \]
- Factor out \( h \):
\[ h (\tan(\phi + \theta) - \tan(\theta)) = a \tan(\theta) \]

9. **Solving for \( h \)**:
- Finally, solve for \( h \):
\[ h = \frac{a \tan(\theta)}{\tan(\phi + \theta) - \tan(\theta)} \]

10. **Final Result**:
- The height of the tower is given by:
\[ h = \frac{a \sin(\theta)}{\sin(\phi) \cos(\theta + \phi)} \]