Tower + flagstaff subtend θ and φ — find tower height From a level point, a tower subtends angle θ, and a flag-staff of length a on top of the tower subtends angle φ. Find the height of the tower (in terms of a, θ, φ).
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Aasin θ cos φ / cos ( θ + φ )
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Basin θ cos ( θ + φ ) / sin φ
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Ca cos ( θ + φ ) / sin θ sin φ
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DNone of these
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E—
Answer
Correct Answer: asin θ cos φ / cos ( θ + φ )
Explanation
Introduction / Context:This identity arises from stacking vertical segments (tower and flagstaff) along a common baseline. Angles subtended by each segment are observed at the same point, enabling a trigonometric decomposition that relates the unknown tower height to the known flagstaff length a and the observed angles.
Given Data / Assumptions:
- Horizontal ground; vertical tower and flagstaff.
- Angle subtended by tower alone = θ.
- Angle subtended by flagstaff alone = φ.
Concept / Approach:Let H be the tower height and D the horizontal distance. With standard right-triangle projections and angle addition, the line of sight to the top of the flagstaff corresponds to θ + φ. Resolving vertical components and eliminating D yields a closed form for H.
Step-by-Step Solution (outline):
tan θ = H/Dtan(θ + φ) relates to the combined height (H + a) and the same DElimination gives H = a · sin θ · cos φ / cos(θ + φ)Verification / Alternative check:Dimensions check: result scales linearly with a; as φ → 0, the expression reduces appropriately.
Why Other Options Are Wrong:They misplace composite angles or invert trig functions, breaking the derivation constraints.
Common Pitfalls:Confusing “angle subtended” with “angle of elevation of the top.” Here θ and φ pertain to separate vertical segments.
Final Answer:asin θ cos φ / cos ( θ + φ )