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Trigonometric Ratios of Compound Angles

A chimney lea...

Question

A chimney leans towards the north. At the points, equal distances due north and south of it in a horizontal plane, the elevations of the tops are $α$ and $β$ respectively. The inclination of the top of the chimney to the vertical is

A

{tan}^{- 1} [\frac{s i n (α - β)}{sin α sin β}]

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B

{tan}^{- 1} [\frac{2 s i n (α - β)}{sin α sin β}]

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C

{tan}^{- 1} [\frac{s i n (α - β)}{2 sin α sin β}]

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D

{tan}^{- 1} [\frac{s i n (α - β)}{2 cos α cos β}]

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Solution

The correct option is C ${tan}^{- 1} [\frac{s i n (α - β)}{2 sin α sin β}]$
OC-chimney and $O A = O B$
Them, according to the $(m - n)$ theorem,'if O be the point on the side AB of a $△ A B C$ which divided the side AB in the ratio $m : n$ , then
$(O A + O B) cot (\frac{π}{2} - θ) = O A cot β - O B cot α$
$(O A + O A) cot (\frac{π}{2} - θ) = O A cot β - O A cot α$
$2 cot (\frac{π}{2} - θ) = cot β - cot α$
$2 tan θ = cot β - cot α [∵ cot (\frac{π}{2} - θ) = tan θ]$
$2 tan θ = \frac{cos β}{sin β} - \frac{cos α}{sin α} = \frac{sin α cos β - cos α sin β}{sin α . sin β}$
$tan θ = \frac{sin (α - β)}{2 sin α sin β}$
$θ = {tan}^{- 1} [\frac{sin (α - β)}{2 sin α sin β}]$

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Similar questions

Q. A chimney leans towards North. At equal distances due North and South of it in a horizontal plane, the elevations of the top are

α, β

. Show that the inclination of the chimney to the vertical is

{tan}^{- 1} [\frac{sin (α - β)}{2 sin α sin β}]

Q. A tree standing on a horizontal plane is leaning towards east. At two points situated at distance

a

b

exactly due west on it, the angles of elevation of the top are respectively

α

β

. Prove that the height of the top from the ground is

\frac{(b - a) tan α tan β}{tan α - tan β}

Q. Two stations due South of a leaning tower which leans towards the North, are at distances

a

b

from its foot. If

α

β

are the elevations of the top of the tower from these stations, then prove that its inclination

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cot θ = \frac{b cot α - a cot β}{b - a}

Q. Question 8
A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height "h" units. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are

α

β

respectively. Prove that the height of the tower is

(\frac{h t a n α}{t a n β - t a n α})

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Q. The elevation of a tower due North of a station

A

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or (at a place due South of it is

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B

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β

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\frac{A B}{\sqrt{({cot}^{2} β - {cot}^{2} α)}}

\frac{A B tan α tan β}{\sqrt{({tan}^{2} α - {tan}^{2} β)}}

\frac{A B sin α sin β}{\sqrt{({sin}^{2} α - {sin}^{2} β)}}

\frac{A B sin α sin β}{{[sin (α + β) sin (α - β)]}^{1 / 2}}

{sin}^{2} α - {sin}^{2} β = sin (α + β) sin (α - β)

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