To the height of a hill CD with its top as C, a horizontal line AB of length a is drawn along the foot of the hill. If `angleCAB = alpha, angleCBA = beta, angleDAC = gamma`, then CD is :

Options

A. `(a sin beta sin gamma)/(sin(alpha + beta))`
B. `(a sin beta sin gamma)/(sin(alpha - beta))`
C. `(sin(alpha - beta))/(a sin beta sin gamma)`
D. `(sin(alpha + beta))/(a sin beta sin gamma)`

Answer

To find the height of the hill \( CD \) given the angles and the length of the horizontal line \( AB \), we can use the properties of triangles and trigonometric functions.

### Step-by-Step Solution:

1. **Identify the Triangle**: We have triangle \( CAB \) where \( C \) is the top of the hill, \( A \) and \( B \) are points on the horizontal line at the base of the hill.

2. **Use the Angles**: From the problem, we know:
- \( \angle CAB = \alpha \)
- \( \angle CBA = \beta \)
- \( \angle DAC = \gamma \)

3. **Apply the Law of Sines**: In triangle \( CAB \), we can use the Law of Sines:
\[ \frac{AB}{\sin C} = \frac{AC}{\sin B} = \frac{BC}{\sin A} \]
where \( C \) is the angle at point \( C \) opposite side \( AB \).

4. **Calculate \( C \)**: Since the angles in a triangle sum up to \( 180^\circ \):
\[ C = 180^\circ - \alpha - \beta \]

5. **Express \( AB \)**: The length of \( AB \) is given as \( a \). Thus, we can write:
\[ \frac{a}{\sin(180^\circ - \alpha - \beta)} = \frac{AC}{\sin \beta} = \frac{BC}{\sin \alpha} \]
Since \( \sin(180^\circ - x) = \sin x \), we have:
\[ \frac{a}{\sin(\alpha + \beta)} = \frac{AC}{\sin \beta} \]

6. **Solve for \( AC \)**: Rearranging gives:
\[ AC = \frac{a \cdot \sin \beta}{\sin(\alpha + \beta)} \]

7. **Use Trigonometry to Find \( CD \)**: The height \( CD \) can be expressed in terms of \( AC \) and the angle \( \gamma \):
\[ CD = AC \cdot \sin(\gamma) \]
Substituting for \( AC \):
\[ CD = \left(\frac{a \cdot \sin \beta}{\sin(\alpha + \beta)}\right) \cdot \sin(\gamma) \]

### Final Expression for \( CD \):
Thus, the height of the hill \( CD \) is given by:
\[ CD = \frac{a \cdot \sin \beta \cdot \sin(\gamma)}{\sin(\alpha + \beta)} \]