First slide

Applications of trigonometry

Question

The distance of three points on a pole from its foot are in A.P. If the angles of elevation of these points from a point on the ground are $α, β$ and $γ$ , then $c o t β c o t γ, c o t γ c o t α a n d c o t α c o t β a r e i n$

Moderate

A

$A . P .$
B

$G . P .$
C

$H . P .$
D

$n o n e$

Solution

$G i v e n A B = a, A C = b, A D = c$
$a, b, c a r e i n A . P . \Rightarrow 2 b = a + c$
$A l s o w e h a v e \tan α = \frac{a}{x}, \tan β = \frac{b}{x}, \tan γ = \frac{c}{x}$
$∴ \cot β . \cot γ + \cot α . \cot β = \frac{x}{b} . \frac{x}{c} + \frac{x}{a} . \frac{x}{b} = \frac{x^{2}}{b} [\frac{a + c}{a c}] = \frac{x^{2}}{b} . \frac{2 b}{a c} = \frac{2 x^{2}}{a c} = 2. \frac{x}{a} . \frac{x}{c} = 2 \cot α . \cot γ$

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