First slide

Applications of trigonometry

Question

A tower subtends angles $α, 2 α$ and $3 α$ respectively at points A, B and C, all lying on a horizontal line through the foot of the tower. Then AB/BC=

Moderate

A

$\frac{\sin 3 α}{\sin 2 α}$
B

$1 + 2 \cos 2 α$
C

$2 + \cos 2 α$
D

$\frac{\sin 2 α}{\sin α}$

Solution

using exterior angle property $∠$ AEB= $α$ and $∠ B E C = α$
using sine rule
$F r o m Δ A B E, \frac{A B}{\sin α} = \frac{B E}{\sin α}, F r o m Δ B C E, \frac{B C}{\sin α} = \frac{B E}{\sin 3 α}$
$\Rightarrow \frac{A B}{B C} = \frac{\sin 3 α}{\sin α} = \frac{3 \sin α - 4 \sin^{3} α}{\sin α}$
$\begin{array}{l} = 3 - 4 \sin^{2} α \\ = 1 + 2 (1 - 2 \sin^{2} α) \\ = 1 + 2 \cos 2 α \end{array}$

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