Question

Solve the question in the attchment​

Answer 1

Question:

The angle of elevation of the top of a tower from a point A due south of the tower is α and from B due east of tower is β. If AB = d, show that the height of the tower is d/√( cot² α+ cot²β )

Answer:

Imagine that you are standing on point A .

From A , let the base of the triangle formed be P .

So AP is the base .

tan α = height / base

Let the height of the tower be h .

⇒ tan α = h / AP

⇒ AP = h / tan α

Similarly in the other triangle formed , the base will be PB .

tan β = height / base

⇒ tan β = h / PB

⇒ PB = h / tan β

Now PB is east and AP is south .

So by Pythagoras theorem :

d² = AP² + PB²

⇒ d² = ( h/tan )² + ( h/tan β )²

⇒ d² = h²cot² + h²cot²β

⇒ d² = h² ( cot² + cot²β )

⇒ h = d/√( cot² α+ cot²β )

Hence proved .

Explanation:

There are 4 cardinal directions - north , south , east , west .

Applying Pythagoras theorem we know :

Hypotenuse² = height² + base²

Answer 2

USE trigonometric relation tan A = height/base

tan α = height / base

⇒ tan alpha= h / x

⇒ x = h / tan alpha

tan β = height / base

⇒ tan β = h / y

⇒ y = h / tan β

d² = x² + y²

= d² = ( h/tan )² + ( h/tan β )²

= d² = h²cot² + h²cot²β

= d² = h² ( cot² + cot²β )

= h = d/√( cot² α+ cot²β )

Hence proved .