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Question

The elevation of a tower due North of a station A is α or (at a place due South of it is α) and at another station B due West of A it is β. Prove that the height of the tower is

AB(cot2βcot2α)

or ABtanαtanβ(tan2αtan2β)

or ABsinαsinβ(sin2αsin2β)

or ABsinαsinβ[sin(α+β)sin(αβ)]1/2 if sin2αsin2β=sin(α+β)sin(αβ)


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Solution

h=PAtanα

PA=hcotα

and h=PBtanβ

PB=hcotβ

Also from the right-angled triangle PAB, we get

PB2=PA2+AB2

AB2=PB2PA2

=h2(cot2βcot2α)

h=AB(cot2βcot2α)
h=ABtanαtanβtan2αtan2β

Substituitng, tanα=sinαcosα

Or ,

h=ABsinαsinβ(sin2αcos2βcos2αsin2β)

Substituting cosβ=(1sin2β) and cosα=(1sin2α)

=ABsinαsinβ[sin2α(1sin2β)sin2β(1sin2α)]
=ABsinαsinβ(sin2αsin2β)

=ABsinαsinβ[sin(α+β)sin(αβ)]

sin2αsin2β=sin(α+β)sin(αβ)

1036509_1006466_ans_c49a0b9025c14d50a2c7b6a4b035f13d.png

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