Question

PQ is post of given height a and AB is a tower at some distance. If alpha and beta are the angles of elevation of B (top of the tower) respectively from PQ. Prove the height AB = a tan alpha / tan alpha - tan beta and the distance between the post and the tower is a / tan alpha - tan beta?

Answer 1

h = height of tower AB 
x = distance from post PQ to tower AB 

tan α = h/x 
x = h / tan α 

tan β = (h-a)/x 
x = (h-a) / tan β 

h / tan α = (h - a) / tan β 
h tan α - a tan α = h tan β 
h tan α - h tan β = a tan α 
h (tan α - tan β) = a tan α 
h = a tan α / (tan α - tan β) 

x = h / tan α 
x = a / (tan α - tan β)

Answer 2

h = height of tower AB 
x = distance from post PQ to tower AB 

tan α = h/x 
x = h / tan α 

tan β = (h-a)/x 
x = (h-a) / tan β 

h / tan α = (h - a) / tan β 
h tan α - a tan α = h tan β 
h tan α - h tan β = a tan α 
h (tan α - tan β) = a tan α 
h = a tan α / (tan α - tan β) 

x = h / tan α 
x = a / (tan α - tan β)