Trignometry (XI Std)?

Question

A man observes a tower at angle @.After walking a distance equal to twice the height of the tower he agins observed the angle to be 2@.What is the value of @?

Ans: 15 (pi/12).I dont know how the answer is obtained.Try as hard as I might I keep on getting @=22.5.

Please HELP.
Thank you.
Tom Marvolo Riddle.

Answer 1

I will show you a different way to solve using trig identities:
Draw the triangle as described with h being the height; "a" is the final distance at an angle of 2x; "a+2h" is the initial distance where angle = x. Since the height is arbitrary set h = 1. We now have:
1/(a+2) = tanx = sinx/cosx
1/a = tanx2x = 2sinxcosx/(cos^2 x - sin^2 x)

solve for a:
sinx/cosx = 2sinxcosx/(cos^2 x + 4sinxcosx - sin^2 x)
cos^2 x = 4sinxcosx - sin^2 x
1 = 4sinxcosx
1/4 = sinxcosx
since sin 2x = 2 sin x cos x
1/2 = sin2x

x = 15deg

Answer 2

Well, draw a picture showing a right triangle ABC with A being where the man starts walking and B being the foot of the tower and C being the top of the tower. Now plac a point D between A and B such that AD = 2BC.

Now angle @ = angle BAC and angle 2@ = angle BDC.
But angle BDC is an exterior angle of triangle and so is = to angle @ + angle ACD. So angle ACD must also = angle @.
Therefore triangle is isosceles and AD = DC
sin 2@ = BC/DC = BC/AD = BC/2BC = 1/2
So 2@ = arcsin(1/2) = 30 degrees.
Thus @= 15 degrees = pi/12 radians

Answer 3

by drawing you will have an isoseles tringle one of its sides =2h where h is the height of the tower and this side is hypotenuse inright triangle in which sin2@=h/2h = 1/2
then 2@= 30 deg.= pi/6 ,then @ = 15deg. =pi/12