Question

The angle \( \phi \) is the angle of elevation from the horizontal to the object. The angle of elevation of the object from the point of observation is \( 1^\circ \). The horizontal distance between the object and the observer is given as \( d \). What is the expression for the distance in terms of \( \phi \)?

• A. \( \cos^2 \phi \)
• B. \( \cos^2 \left( \frac{\phi}{2} \right) \)
• C. \( \sin^2 \phi \)
• D. \( \sin^2 \left( \frac{\phi}{2} \right) \)

Hint:

When dealing with angles of elevation, always check the trigonometric identities that connect the angle with the distances, especially when angles involve divisions such as \( \frac{\phi}{2} \).

Answer 1

The Correct Option is B

Step 1: Understanding the scenario.
In trigonometry, the distance between the observer and the object can be related to the angle of elevation and the horizontal distance. The relationship between these quantities involves the cosine of half the angle in the context of angles involving \( 1^\circ \).

Step 2: Applying the formula.
Given that the horizontal distance between the object and the observer is \( d \), the correct expression for the distance, based on standard trigonometric relationships, is: \[ d = \cos^2 \left( \frac{\phi}{2} \right) \]
Step 3: Conclusion.
The correct answer is (B) because it represents the proper trigonometric identity for the given angle and distance relation.