First slide

Applications of trigonometry

Question

A flag staff $A B$ 4m high is placed on a vertical tower. The angle of elevation of foot of the flag staff $B$ at two points $D, E$ on the ground level and lying on either side of foot of the tower are $α$ and 60⁰ respectively with 4m and $2 \sqrt{3} m$ as the respective distances of these points from the foot of tower. The angle subtended by the flag staff at point $D$ is

Moderate

A

$\tan^{- 1} \frac{2}{19}$
B

$\tan^{- 1} \frac{2}{9}$
C

$\tan^{- 1} \frac{19}{2}$
D

$\tan^{- 1} \frac{4}{19}$

Solution

$\begin{array}{l} From Δ B C E, \cot 60^{\circ} = \frac{2 \sqrt{3}}{x} \\ \Rightarrow x = 6 \\ From △ A C D, \tan (θ + α) = \frac{10}{4} = \frac{5}{2}; \tan α = \frac{3}{2} \\ \Rightarrow \frac{\tan θ + \frac{3}{2}}{1 - \tan θ \cdot \frac{3}{2}} = \frac{5}{2} (∵ \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}) \end{array}$
$\begin{array}{l} \Rightarrow t a n θ = \frac{4}{19} \\ \Rightarrow θ = \tan^{- 1} (\frac{4}{19}) \end{array}$
Therefore, the correct answer is (4).

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