Example 13. Two stations due south of a leaning tower which leads towards the north are at
distances a and ) from its foot. I o, be the elevations of the top of the tower from these stations,
b cot a-a cot B
prove that its inclination to the horizontal is given by cot e =
b-a
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10th
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Some Applications of Trigonometry
Heights and Distances
Two stations due South of a...
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Asked on November 22, 2019 by
Yuraj Tehreem
Two stations due South of a leaning tower which leans towards the North, are at distances a and b from its foot. If α and β are the elevations of the top of the tower from these stations, then prove that its inclination θ to the horizontal is given by
cotθ=
b−a
bcotα−acotβ
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ANSWER
height of the tower DE=h
Distance between first station to foot of tower AD=a+x
Distance between second station to foot of tower BD=b+x
Distance between CandD=x
Give α, β are the angle of elevation two stations to top of the tower
thatis∠DAE=α,∠DBE=β,∠DCE=θ
InΔADE
Cotθ=x/h−−−−−−−−−−−−−−−−→(1)
InΔBDE
Cotβ=(b+x)/h
(b+x)=hCotβ (multiply a on both sides )
(ab+ax)=haCotβ−−−−−−−−−−−−−−−−→(2)
InΔCDE
Cotα=(a+x)/h
(a+x)=hCotα (multiply b on both sides )
(ab+bx)=hbCotα−−−−−−−−−−−−−−−−→(3)
substract(3)−(2)
(b−a)x=h(bCotα−aCotβ)
x/h=(bCotα−aCotβ)/(b−a)
Cotθ=(bCotα−aCotβ)/(b−a).
Answer:
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