The angle of elevation of the top of a certain tower from a point A on the ground is `alpha`, at B is `2alpha`, at C is `3alpha`. If `AB= 4/3 BC`, then which of the following is true.

Options

A. `sin alpha =sqrt(5/12)`
B. `cos alpha = sqrt(5/12)`
C. `sin alpha = 3/4`
D. `cos alpha = 3/8`

Answer

To solve the problem step-by-step, we will analyze the situation using trigonometric concepts and the properties of triangles.

### Step 1: Understand the Problem
We have a tower of height \( H \) and three points \( A \), \( B \), and \( C \) on the ground. The angles of elevation from these points to the top of the tower are \( \alpha \), \( 2\alpha \), and \( 3\alpha \) respectively. We are also given that \( AB = \frac{4}{3} BC \).

### Step 2: Set Up the Diagram
Draw a diagram with the tower \( PQ \) and the points \( A \), \( B \), and \( C \) on the ground. The angles of elevation can be represented as follows:
- From point \( A \): angle \( \angle PAQ = \alpha \)
- From point \( B \): angle \( \angle PBQ = 2\alpha \)
- From point \( C \): angle \( \angle PCQ = 3\alpha \)

### Step 3: Apply Trigonometric Ratios
Using the definition of sine in right triangles, we can express the distances \( AP \) and \( PC \):
- From triangle \( APQ \):
\[ \sin(\alpha) = \frac{H}{AP} \implies AP = \frac{H}{\sin(\alpha)} \]
- From triangle \( PCQ \):
\[ \sin(3\alpha) = \frac{H}{PC} \implies PC = \frac{H}{\sin(3\alpha)} \]

### Step 4: Use the Given Ratio
We know that:
\[ \frac{AB}{BC} = \frac{AP}{PC} = \frac{4}{3} \]
Let \( BC = x \). Then \( AB = \frac{4}{3}x \).

### Step 5: Express \( AB \) and \( BC \) in Terms of \( H \)
Using the distances:
\[ AB = AP - BP \quad \text{and} \quad BC = PC - PB \]
From the triangle properties, we can express \( BP \) and \( PB \) in terms of \( H \) and the angles:
- \( BP = AP - AB \)
- \( PB = PC - BC \)

### Step 6: Substitute and Simplify
Using the ratios:
\[ \frac{AP}{PC} = \frac{4}{3} \implies \frac{\frac{H}{\sin(\alpha)}}{\frac{H}{\sin(3\alpha)}} = \frac{4}{3} \]
This simplifies to:
\[ \frac{\sin(3\alpha)}{\sin(\alpha)} = \frac{4}{3} \]

### Step 7: Apply the Sine Triple Angle Formula
Using the formula for \( \sin(3\alpha) \):
\[ \sin(3\alpha) = 3\sin(\alpha) - 4\sin^3(\alpha) \]
Substituting this into our equation gives:
\[ \frac{3\sin(\alpha) - 4\sin^3(\alpha)}{\sin(\alpha)} = \frac{4}{3} \]
This simplifies to:
\[ 3 - 4\sin^2(\alpha) = \frac{4}{3} \]

### Step 8: Solve for \( \sin^2(\alpha) \)
Multiply through by 3 to eliminate the fraction:
\[ 9 - 12\sin^2(\alpha) = 4 \]
Rearranging gives:
\[ 12\sin^2(\alpha) = 5 \implies \sin^2(\alpha) = \frac{5}{12} \]

### Step 9: Final Result
Taking the square root gives:
\[ \sin(\alpha) = \sqrt{\frac{5}{12}} \]

### Conclusion
Thus, the required answer is \( \sin(\alpha) = \sqrt{\frac{5}{12}} \).