The angle of elevation of the top of tower from a point A due south of it is 68 degrees .From a point B due east of A the angle of elevation of the top is 54 degrees. If A is 50 m from B find the height of the tower.
Haven't you tried anything yourself? Start by drawing a picture. There are four triangles involved. Two are the right triangles formed by
1) the base of the tower, the top of the tower, and point A. It has angle 68 degrees.
2) the base of the tower, the top of the tower, and point B. It has angle 54 degrees.
There are two other triangles formed by
3) the base of the tower, point A, and point B. The side between points A and B is 50 m long.
4) the top of the tower, point A, and point B.
Let T be the top of the tower, rectangle ABCD where D is the foot of the tower so that
AB = CD =50 , angle TAD = 68 and angle TBD = 54. Let AD = BC = d and TD = h.
Let angle TBC = x and angle DBC = y so that x - y = 54
$\displaystyle \tan x - \tan y = \frac{h+50}{d}-\frac{50}{d}=\frac{h}{d}=\tan 68$
and
$\displaystyle \tan x \tan y = \frac{(h+50)50}{d^2}=\frac{(h+50)50}{h^2/\tan ^2 68}$
Replace these quantities in the identity
$\displaystyle \tan 54 = \tan (x-y)=\frac{\tan x - \tan y}{1+\tan x \tan y}$
to get a quadratic equation in $\displaystyle h$
$\displaystyle h=428.49$