Question

A ladder rests against a wall at an angle α to the horizontal. Its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle β with the horizontal. Show that a/b=cosα-cosβ/sinβ-sinα

Answer 1

a/b = (Cosα - Cosβ  )/(Sinβ - Sinα)

Step-by-step explanation:

Let say Ladder length = L

A ladder rests against a wall at an angle α to the horizontal.

=> Cosα  = Distance of Ladder from wall / Length of Ladder

=> Distance of Ladder from wall = L Cosα

=> Sinα  = Height of Ladder from wall / Length of Ladder

=> Height of Ladder from wall = L Sinα

Its foot is pulled away from the wall through a distance a

Cosβ  = (L Cosα + a)/L

=> LCosβ = L Cosα + a

=> L (Cosβ - Cosα ) = a

=> L = a/(Cosβ - Cosα )

Sinβ  = (L Sinα - b)/L

=> LSinβ = L Sinα - b

=> b = L (Sinα - Sinβ)

=> L = b/(Sinα - Sinβ)

a/(Cosβ - Cosα ) =  b/(Sinα - Sinβ)

=> a/b = (Cosβ - Cosα ) /(Sinα - Sinβ)

=> a/b = -(Cosα - Cosβ  )/(-(Sinβ - Sinα))

=> a/b = (Cosα - Cosβ  )/(Sinβ - Sinα)

QED

Proved

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