A vertical tower stands at a point P within the horizontal triangle ABC and makes angles - - - at A- B and C respectively- If the sides of the triangle makes equal angles at P- then prove that - sin2 A cot - - cot - - 0

Sides subtend equal angles at P which is 120^o each. PA = h α, PB = h β, PC = h γ From Δ PBC by cosine rule. BP^2 + CP^2 BC^22 BP. CP = co ...

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Standard XII

Mathematics

Basic Inverse Trigonometric Functions

A vertical to...

Question

A vertical tower stands at a point P within the horizontal triangle ABC and makes angles $α, β, γ$ at A, B and C respectively. If the sides of the triangle makes equal angles at P, then prove that
$\sum s i n^{2} A (c o t β - c o t γ) = 0$

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Solution

Sides subtend equal angles at P which is $120^{o}$ each.
$P A = h cot α, P B = h cot β, P C = h cot γ$
From $Δ P B C$ by cosine rule.
$\frac{B P^{2} + C P^{2} - B C^{2}}{2 B P . C P} = cos ∠ B P$
$= cos 120^{o} = \frac{1}{2}$
$h^{2} {cot}^{2} β + h^{2} {cot}^{2} γ = a^{2} = - h^{2} cot β cot γ$
or $h^{2} ({cot}^{2} β + {cot}^{2} γ + cot β cot γ) = a^{2}$
Multiply both sides by $cot β - cot γ$ , we get
$h^{2} ({cot}^{3} β - {cot}^{3} γ) = a^{2} (cot β - cot γ)$
Write similar expressions and add
$h^{2} . (0) = \sum a^{2} (cot β - cot γ)$
or $\sum {sin}^{2} A (cot β - cot γ) = 0$

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A vertical tower stands at a point P within the horizontal triangle ABC and makes angles α,β,γ at A, B and C respectively. If the sides of the triangle makes equal angles at P, then prove that∑sin2A(cotβ−cotγ)=0

A vertical tower stands at a point P within the horizontal triangle ABC and makes angles $α, β, γ$ at A, B and C respectively. If the sides of the triangle makes equal angles at P, then prove that
$\sum s i n^{2} A (c o t β - c o t γ) = 0$