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 1: Define the Variables
Let:
- \( h \) = height of the tower
- \( a \) = height of the flagstaff
- \( \theta \) = angle subtended by the tower at the point
- \( \phi \) = angle subtended by the flagstaff at the same point

### Step 2: Set Up the Right Triangles
From the point on the level plane:
- The angle \( \theta \) subtended by the tower gives us the relation:
\[ \tan(\theta) = \frac{h}{d} \]
where \( d \) is the horizontal distance from the point to the base of the tower.

- The angle \( \phi \) subtended by the flagstaff gives us:
\[ \tan(\phi) = \frac{h + a}{d} \]

### Step 3: Express \( d \) in Terms of \( h \) and \( a \)
From the first equation, we can express \( d \) as:
\[ d = \frac{h}{\tan(\theta)} \]

Substituting this into the second equation:
\[ \tan(\phi) = \frac{h + a}{\frac{h}{\tan(\theta)}} \]

### Step 4: Simplify the Equation
Rearranging the equation gives:
\[ \tan(\phi) = \frac{(h + a) \tan(\theta)}{h} \]

Cross-multiplying:
\[ h \tan(\phi) = (h + a) \tan(\theta) \]

### Step 5: Expand and Rearrange
Expanding the right side:
\[ h \tan(\phi) = h \tan(\theta) + a \tan(\theta) \]

Rearranging gives:
\[ h \tan(\phi) - h \tan(\theta) = a \tan(\theta) \]

Factoring out \( h \):
\[ h (\tan(\phi) - \tan(\theta)) = a \tan(\theta) \]

### Step 6: Solve for \( h \)
Now, we can solve for \( h \):
\[ h = \frac{a \tan(\theta)}{\tan(\phi) - \tan(\theta)} \]

### Step 7: Substitute Trigonometric Identities
Using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \):
\[ h = \frac{a \frac{\sin(\theta)}{\cos(\theta)}}{\frac{\sin(\phi)}{\cos(\phi)} - \frac{\sin(\theta)}{\cos(\theta)}} \]

### Step 8: Simplify Further
This can be simplified to:
\[ h = a \cdot \frac{\sin(\theta) \cos(\phi)}{\sin(\phi) \cos(\theta) - \sin(\theta) \cos(\phi)} \]

### Final Expression
Thus, the height of the tower \( h \) can be expressed as:
\[ h = a \cdot \frac{\sin(\theta) \cos(\phi)}{\sin(\phi) \cos(\theta) - \sin(\theta) \cos(\phi)} \]