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 need to find the height of the TV tower given the angles of elevation from three points A, B, and C. Let's denote the height of the tower as \( H \) and the distances from the foot of the tower to points A, B, and C as follows:

1. Let the distance from the foot of the tower to point A be \( AQ = a + y \).
2. Let the distance from the foot of the tower to point B be \( BQ = y \).
3. The angle of elevation from point A is \( \alpha \), from point B is \( 2\alpha \), and from point C is \( 3\alpha \).

### Step 1: Establishing the relationships using trigonometry

From triangle \( APQ \):
\[ \tan(\alpha) = \frac{H}{AQ} = \frac{H}{a + y} \]
Thus, we can express \( H \) as:
\[ H = (a + y) \tan(\alpha) \quad \text{(1)} \]

From triangle \( PBQ \):
\[ \tan(2\alpha) = \frac{H}{BQ} = \frac{H}{y} \]
Thus, we can express \( H \) as:
\[ H = y \tan(2\alpha) \quad \text{(2)} \]

### Step 2: Setting the equations equal

From equations (1) and (2), we can set them equal to each other:
\[ (a + y) \tan(\alpha) = y \tan(2\alpha) \]

### Step 3: Expanding and rearranging the equation

Expanding the left side:
\[ a \tan(\alpha) + y \tan(\alpha) = y \tan(2\alpha) \]
Rearranging gives:
\[ a \tan(\alpha) = y \tan(2\alpha) - y \tan(\alpha) \]
Factoring out \( y \) from the right side:
\[ a \tan(\alpha) = y (\tan(2\alpha) - \tan(\alpha)) \]

### Step 4: Solving for \( y \)

Now, we can solve for \( y \):
\[ y = \frac{a \tan(\alpha)}{\tan(2\alpha) - \tan(\alpha)} \quad \text{(3)} \]

### Step 5: Finding \( H \) using \( y \)

Substituting equation (3) back into equation (2) to find \( H \):
\[ H = y \tan(2\alpha) = \frac{a \tan(\alpha)}{\tan(2\alpha) - \tan(\alpha)} \tan(2\alpha) \]

### Step 6: Simplifying \( H \)

Now we can simplify \( H \):
\[ H = \frac{a \tan(\alpha) \tan(2\alpha)}{\tan(2\alpha) - \tan(\alpha)} \]

### Step 7: Using the identity for \( \tan(2\alpha) \)

Using the identity for \( \tan(2\alpha) = \frac{2 \tan(\alpha)}{1 - \tan^2(\alpha)} \):
\[ H = \frac{a \tan(\alpha) \cdot \frac{2 \tan(\alpha)}{1 - \tan^2(\alpha)}}{\frac{2 \tan(\alpha)}{1 - \tan^2(\alpha)} - \tan(\alpha)} \]

### Final Step: Arriving at the final formula for \( H \)

After simplification, we find:
\[ H = a \sin(2\alpha) \]

Thus, the height of the tower is:
\[ \boxed{a \sin(2\alpha)} \]