First slide

Applications of trigonometry

Question

A tower, of x meters high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant ‘y’ meters from the foot of the tower. Then the length of the flagstaff in meters is

Moderate

A

$\frac{y (x^{2} - y^{2})}{(x^{2} + y^{2})}$
B

$\frac{x (x^{2} + y^{2})}{(y^{2} - x^{2})}$
C

$\frac{x (x^{2} + y^{2})}{(x^{2} - y^{2})}$
D

$\frac{x (x^{2} - y^{2})}{(x^{2} + y^{2})}$

Solution

$W e h a v e \tan θ = \frac{x}{y}, \tan (2 θ) = \frac{h + x}{y},$
$\Rightarrow h + x = y \tan 2 θ = y \times \frac{2 \tan θ}{1 - \tan^{2} θ} = y \frac{2 x / y}{1 - x^{2} / y^{2}} = \frac{2 x y^{2}}{y^{2} - x^{2}}$
$\Rightarrow h = \frac{2 x y^{2}}{y^{2} - x^{2}} - x = \frac{2 x y^{2} - x y^{2} + x^{3}}{y^{2} - x^{2}} = \frac{x (y^{2} + x^{2})}{(y^{2} - x^{2})}$

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