From a window x meters high above the ground in a street, the angles of elevation and depression of the top and the foot of the other house on the opposite side of the street are α and β respectively. Show that the height of the opposite house is x(1+ tanα cotβ)Meters.
Ans: Let AB be the house and P be the window
Let BQ = x ∴ PC = x
Let AC = a
In Δ PQB, tan θ = PQ/QB or tan θ = h/x
∴ x = h/tan θ = h cot θ
In Δ PAC, tan θ = AC/PC or tan θ = a/x
∴ a = x tan θ > (h cot θ) tan θ = h tan θ cot θ.
∴ the height of the tower = AB = AC + BC
= a + h = h tan θ cot θ + h = h (tanθ cot θ + 1)