Question

From the bottom of a pole of height h, the angle of elevation of the top of a tower is \( \alpha \). The pole subtends an angle \( \beta \) at the top of the tower. The height of the tower is

• A. \[ \frac{h \sin \alpha \sin (\alpha - \beta)}{\sin \beta} \]
• B. \[ \frac{h \sin \alpha \cos (\alpha + \beta)}{\cos \beta} \]
• C. \[ \frac{h \sin \alpha \cos (\alpha - \beta)}{\sin \beta} \]
• D. \[ \frac{h \sin \alpha \sin (\alpha + \beta)}{\cos \beta} \]

Hint:

For problems involving angles and heights, use trigonometric identities to form relations between the given angles and known values.

Answer 1

The Correct Option is C

We are given that the height of the pole is \( h \), and the angle of elevation of the top of the tower is \( \alpha \).
The angle subtended by the pole at the top of the tower is \( \beta \).
To find the height of the tower, we can use the following approach: 
1. Using the relation between the angles and height, we can set up a geometric relation based on trigonometry.
2. Since the angle of elevation is given as \( \alpha \), the height of the tower is related to the angle \( \beta \) and the height of the pole through the formula: \[ \text{Height of the tower} = \frac{h \sin \alpha \cos (\alpha - \beta)}{\cos \beta} \]