From the top of a cliff of height 'a' the angle off depression of the foot of a certain tower is found to be double the angle of elevation of the top of the tower of height h. if `theta` be the angle of elevation then its value is

Options

A. `cos^(-1)sqrt((2h)/(a))`
B. `sin^(-1)sqrt((2h)/(a))`
C. `sin^(-1)sqrt((a)/(2-h))`
D. `tan^(-1)sqrt(3-(2h)/(a))`

Answer

To solve the problem, we need to analyze the situation involving a cliff and a tower, and use trigonometric relationships to find the angle of elevation, θ.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the height of the cliff be \( A \).
- Let the height of the tower be \( H \).
- The angle of elevation from the top of the cliff to the top of the tower is \( \theta \).
- The angle of depression from the top of the cliff to the foot of the tower is \( 2\theta \).

2. **Drawing the Diagram**:
- Draw a vertical line representing the cliff of height \( A \).
- Draw a vertical line next to it representing the tower of height \( H \).
- Mark the point at the top of the cliff as point \( P \) and the foot of the tower as point \( Q \).
- The angle of elevation \( \theta \) is formed at point \( P \) looking towards the top of the tower, and the angle of depression \( 2\theta \) is formed at point \( P \) looking down towards the foot of the tower.

3. **Using Trigonometric Ratios**:
- From point \( P \) to point \( Q \), the horizontal distance can be denoted as \( x \).
- For the angle of elevation \( \theta \):
\[ \tan(\theta) = \frac{H - A}{x} \quad \text{(1)} \]
- For the angle of depression \( 2\theta \):
\[ \tan(2\theta) = \frac{A}{x} \quad \text{(2)} \]

4. **Using the Double Angle Formula**:
- We know from trigonometry that:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
- Substituting the expressions from equations (1) and (2):
\[ \frac{A}{x} = \frac{2\frac{H - A}{x}}{1 - \left(\frac{H - A}{x}\right)^2} \]

5. **Cross-Multiplying and Simplifying**:
- Cross-multiplying gives:
\[ A(1 - \left(\frac{H - A}{x}\right)^2) = 2(H - A) \]
- Expanding and simplifying leads to:
\[ A - \frac{A(H - A)^2}{x^2} = 2H - 2A \]
- Rearranging gives:
\[ A + 2A = 2H + \frac{A(H - A)^2}{x^2} \]
- This simplifies to:
\[ 3A = 2H + \frac{A(H - A)^2}{x^2} \]

6. **Finding the Value of θ**:
- Rearranging the equation leads to a relationship between \( H \), \( A \), and \( \theta \):
\[ \tan^2(\theta) = \frac{3 - \frac{2H}{A}}{2} \]
- Finally, we can express \( \theta \) as:
\[ \theta = \tan^{-1}\left(\sqrt{\frac{3 - \frac{2H}{A}}{2}}\right) \]

### Final Result:
Thus, the value of \( \theta \) is:
\[ \theta = \tan^{-1}\left(\sqrt{\frac{3 - \frac{2H}{A}}{2}}\right) \]