Trigonometry
Heights and Distances
Grade Class 12
Question:
A tower is observed from three collinear points A, B, C on ground such that angles of elevation are $\alpha$, $2\alpha$, $3\alpha$. If $AB : BC = ?$
Step-by-Step Solution
Key Concept: Use the trigonometric form for collinear observation points; $\frac{AB}{BC} = \frac{\sin(3\alpha-2\alpha)\sin 2\alpha}{\sin(2\alpha-\alpha)\sin 3\alpha}$ from sine rule in triangles.
Let tower height $h$, tower base $T$. Using $\cot\alpha - \cot 2\alpha = AT/h$ etc.: $\frac{AB}{BC}=\frac{\cot\alpha-\cot 2\alpha}{\cot 2\alpha-\cot 3\alpha}$. After simplification using product-to-sum identities, $\frac{AB}{BC} = 1+2\cos 2\alpha$.
Correct Answer: 2