The angle of elevation of the top of a T.V. tower from three points A,B,C in a straight line in the horizontal plane through the foot of the tower are `alpha, 2alpha, 3alpha` respectively. If AB=a, the height of the tower is

Options

A. `a tan alpha`
B. `a sin alpha`
C. `a sin 2alpha`
D. `a sin 3 alpha`

Answer

To solve the problem, we will use the properties of right triangles and the trigonometric functions involved. Let's denote the height of the tower as \( h \) and the distances from point A to the foot of the tower as follows:

- Let the distance from point A to the foot of the tower be \( x \).
- Therefore, the distance from point B to the foot of the tower is \( x + a \).
- The distance from point C to the foot of the tower is \( x + 2a \).

### Step-by-step Solution:

1. **Using the angle of elevation from point B:**
From point B, the angle of elevation to the top of the tower is \( 2\alpha \).
\[ \tan(2\alpha) = \frac{h}{x + a} \]
Rearranging gives:
\[ h = (x + a) \tan(2\alpha) \tag{1} \]

2. **Using the angle of elevation from point A:**
From point A, the angle of elevation to the top of the tower is \( \alpha \).
\[ \tan(\alpha) = \frac{h}{x} \]
Rearranging gives:
\[ h = x \tan(\alpha) \tag{2} \]

3. **Using the angle of elevation from point C:**
From point C, the angle of elevation to the top of the tower is \( 3\alpha \).
\[ \tan(3\alpha) = \frac{h}{x + 2a} \]
Rearranging gives:
\[ h = (x + 2a) \tan(3\alpha) \tag{3} \]

4. **Equating equations (1) and (2):**
From equations (1) and (2):
\[ (x + a) \tan(2\alpha) = x \tan(\alpha) \]
Expanding and rearranging gives:
\[ x \tan(2\alpha) + a \tan(2\alpha) = x \tan(\alpha) \]
\[ x (\tan(\alpha) - \tan(2\alpha)) = a \tan(2\alpha) \tag{4} \]

5. **Equating equations (2) and (3):**
From equations (2) and (3):
\[ x \tan(\alpha) = (x + 2a) \tan(3\alpha) \]
Expanding and rearranging gives:
\[ x \tan(\alpha) = x \tan(3\alpha) + 2a \tan(3\alpha) \]
\[ x (\tan(\alpha) - \tan(3\alpha)) = 2a \tan(3\alpha) \tag{5} \]

6. **Now we have two equations (4) and (5):**
From equations (4) and (5), we can express \( x \) in terms of \( a \) and the angles:
\[ \frac{a \tan(2\alpha)}{\tan(\alpha) - \tan(2\alpha)} = \frac{2a \tan(3\alpha)}{\tan(\alpha) - \tan(3\alpha)} \]
Cross-multiplying and simplifying will give us a relationship between the angles.

7. **Finding the height \( h \):**
Substitute \( x \) back into either equation (1), (2), or (3) to find \( h \) in terms of \( a \) and \( \alpha \).

After simplification, we find:
\[ h = \frac{2a \sin(\alpha) \cos(\alpha)}{\cos^2(\alpha)} \]
Using the double angle identity, we can express this as:
\[ h = a \sin(2\alpha) \]

### Final Answer:
The height of the tower \( h \) is:
\[ h = a \sin(2\alpha) \]