Question:

From the top of a lighthouse, the angles of depression of two stations on the oposite sides of it at a distance d apart are $\alpha$ and $\beta$. The height of the lighthouse is

Updated On: Jun 20, 2022
  • $\frac{d \, \tan \, \alpha}{\tan \, \alpha + \tan \, \beta}$
  • $\frac{d}{\cot \, \alpha + \cot \, \beta}$
  • $\frac{d \, \tan \, \beta}{\tan \, \alpha + \tan \, \beta}$
  • $\frac{d \, \cot \, \beta}{\cot \, \alpha + \cot \, \beta}$
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The Correct Option is B

Solution and Explanation

Let $PM$ be lighthouse.



Given,
$\angle P Q M=\alpha, \angle P R M=\beta$ and $Q R=d$
In $\Delta PQM$,
$tan\,\alpha=\frac{PM}{QM}$
$\Rightarrow QM=PM\,cot\,\alpha$
In $\Delta PRM$,
$\tan\,\beta=\frac{PM}{RM}$
$\Rightarrow RM = PM\,\cot\,??
$\therefore QR=QM+MR$
$\Rightarrow d=PM\,cot\,\alpha+PM\, \cot\,\beta$
$\Rightarrow d=PM\left(\cot\,\alpha + \cot\,\beta\right)$
$\Rightarrow d=PM\left(\cot\,\alpha + \cot\,\beta\right)$
$\Rightarrow PM=\frac{d}{\cot\,\alpha + \cot\,\beta}$
$\therefore$ Height of light house is $\frac{d}{\cot\,\alpha + \cot\,\beta}$.
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Concepts Used:

Trigonometric Identities

Various trigonometric identities are as follows:

Even and Odd Functions

Cosecant and Secant are even functions, all the others are odd.

  • sin (-A) = – sinA,
  • cos (-A) = cos A,
  • cosec (-A) = -cosec A,
  • cot (-A) = -cot A,
  • tan (-A) = – tan A,
  • sec (-A) = sec A.

Pythagorean Identities

  1. sin2θ + cos2θ = 1
  2. 1 + tan2θ = sec2θ
  3. 1 + cot2θ = cosec2θ

Periodic Functions

  1. T-Ratios of (2π + x)
    sin (2π + x) = sin x,
    cos (2π + x) = cos x,
    tan (2π + x) = tan x,
    cosec (2π + x) = cosec x,
    sec (2π + x) = sec x,
    cot (2π+x)=cotx.
  2. T-Ratios of (π -x)
    sin (π–x) = sin x,
    cos (π–x) = - cos x,
    tan (π–x) = - tan x,
    cosec (π–x) = cosec x,
    sec (π–x) = - sec x,
    cot (π–x) = - cot x.
  3. T-Ratios of (π+ x)
    sin (π+x) = - sin x,
    cos (π+x) = - cos x,
    tan (π+x) = tan x,
    cosec (π+x) = - cosec x,
    sec (π+x) = - sec x,
    cot (π+x) = cot x.
  4. T-Ratios of (2π – x)
    sin (2π–x) = - sin x,
    cos (2n–x) = cos x,
    tan (2π–x) = - tan x,
    cosec (2π–x) = - cosec x,
    sec (2π–x) = sec x,
    cot (2π-x) = - cot x

Sum and Difference Identities

  1. T-Ratios of (x + y)
    sin (x+y) = sinx.cosy + cosx.sin y
    cos (x+y) = cosx.cosy – sinx.siny
  2. T-Ratios of (x – y)
    sin (x–y) = sinx.cosy – cos.x.sin y
    cos (x-y) = cosx.cosy + sinx.siny

Product of T-ratios

  • 2sinx cosy = sin(x+y) + sin(x–y)
  • 2cosx siny = sin(x+y) – sin(x–y)
  • 2 cosx cosy = cos(x+y) + cos(x–y)
  • 2sinx.siny = cos(x–y) – cos(x+y)

T-Ratios of (2x)
sin2x = 2sin x cos x
cos 2x = cos2x – sin2x 

= 2cos2x – 1 

= 1 – 2sin2x

T-Ratios of (3x)
sin 3x = 3sinx – 4sin3x
cos 3x = 4cos3x – 3cosx