Question:

A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min. for the angle of depression of the car to change from $30^{\circ}$ to $45^{\circ}$ ; then after this, the time taken (in min.) by the car to reach the foot of the tower, is :

Updated On: Jul 15, 2024
  • $9( 1 +\sqrt{3})$
  • $ 18 ( 1 +\sqrt{3})$
  • $18 ( \sqrt{3} - 1) $
  • $\frac{9}{2} ( \sqrt{3} - 1) $
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The Correct Option is A

Solution and Explanation

Let length of tower = h $\Rightarrow AC' = AB = h$ and $AC = AB cot 30^{\circ}= \sqrt{3} h$ $\Rightarrow CC' = (\sqrt{3} -1) h $ Time taken by car form C to C' = 18 min $\Rightarrow$ time take by car to reach the foot of the tower = $\frac{18}{\sqrt{3}-1}$ min. = 9 ( $\sqrt{3}$ + 1) min
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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