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 given information and apply trigonometric principles to find the relationship between the angles and distances involved.

### Step 1: Understand the Setup
We have three points A, B, and C on the ground, with angles of elevation to the top of the tower being α, 2α, and 3α respectively. The distance between points A and B is given as \( AB = \frac{4}{3} BC \).

### Step 2: Use the Exterior Angle Theorem
From point B, the angle of elevation is 2α. According to the exterior angle theorem, we can express the angle at point B (angle ABC) as:
\[ \angle ABC = \angle A + \angle ACB \implies 2α = α + \angle ACB \implies \angle ACB = α \]

Similarly, for point C, we can apply the exterior angle theorem:
\[ \angle BPC = 3α = 2α + \angle ACB \implies \angle ACB = α \]

### Step 3: Establish the Angle Bisector
Since both calculations show that angle ACB = α, we conclude that line PB bisects angle ABC. Therefore, by the angle bisector theorem, we can say:
\[ \frac{AB}{BC} = \frac{AP}{PC} \]

### Step 4: Calculate AP and PC
Using trigonometric ratios in the respective triangles, we can express AP and PC in terms of the height of the tower (h) and the angles:
- For triangle ABP:
\[ AP = \frac{h}{\sin(α)} \]
- For triangle BCP:
\[ PC = \frac{h}{\sin(3α)} \]

### Step 5: Substitute AP and PC into the Ratio
Now, substituting these values into the ratio we established:
\[ \frac{AB}{BC} = \frac{AP}{PC} = \frac{\frac{h}{\sin(α)}}{\frac{h}{\sin(3α)}} = \frac{\sin(3α)}{\sin(α)} \]

### Step 6: Set Up the Equation
From the problem, we know:
\[ AB = \frac{4}{3} BC \implies \frac{AB}{BC} = \frac{4}{3} \]
Thus, we can equate:
\[ \frac{\sin(3α)}{\sin(α)} = \frac{4}{3} \]

### Step 7: Use the Identity for sin(3α)
Using the trigonometric identity for sin(3α):
\[ \sin(3α) = 3\sin(α) - 4\sin^3(α) \]
Substituting this into our equation gives:
\[ \frac{3\sin(α) - 4\sin^3(α)}{\sin(α)} = \frac{4}{3} \]
Simplifying this leads to:
\[ 3 - 4\sin^2(α) = \frac{4}{3} \]

### Step 8: Solve for sin²(α)
Multiplying through by 3 to eliminate the fraction:
\[ 9 - 12\sin^2(α) = 4 \implies 12\sin^2(α) = 5 \implies \sin^2(α) = \frac{5}{12} \]

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

### Conclusion
Thus, the final result is:
\[ \sin(α) = \sqrt{\frac{5}{12}} \]