Show that : a/b=cos alpha-cos beta/sin beta-sin alpha

Question

A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slids down a distance  b down the wall making an angle ß with the horizontal.

Show that: a/b=(cos alpha-cos ß)/(sin ß-sin alpha)

Answer 1

Let A=alpha and B=beta for convenience.

When the ladder is moved its vertical height changes from LsinA to LsinA-b.

Its horizontal distance from the wall changes from LcosA to LcosA+a.

Let L=length of ladder. Considering the position after the ladder is moved we can write:

LsinA-b=LsinB and LcosA+a=LcosB.

Therefore, a=L(cosB-cosA) and b=L(sinA-sinB), so

a/b=(cosB-cosA)/(sinA-sinB) or (cosA-cosB)/(sinB-sinA).