Trigonometry-Height& distance '10points'!!!!?

Question

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

a/b=(cos alpha-cos beta)/(sin beta-sin alpha)

Please mention all the steps and also the diagram.

I am unable to draw the diagram correctly.

I will certainly give 10 points to one who gives correct answer .

Answer 1

first you draw a right angled triangle PQR with angle Q = 90 degrees and PQ vertical and QR horizontal to your right.

Join ST where T lies on QR produced with RT = a and S lies on PQ between P and Q with PS = b.

In both triangles PQR and SQT, ladder is PR = ST

and angle PRQ = alpha, angle STQ = beta.

whence

(cos alpha-cos beta)/(sin beta-sin alpha)

= {QR/PR -- QT/ST} / {SQ/ST -- PQ/PR}

= (QR -- QT) / (SQ -- PQ) because PR = ST

= (QR -- QR -- RT) / (SQ -- SQ -- PS)

= (--RT) / (--PS)

= RT / PS

= a / b  

Answer 2

Diagram:

-Draw a vertical wall and a horizontal ground. -Draw Ladder 1 leaning against the wall and the ground. The angle it makes with the ground is alpha.

-Draw Ladder 2 leaning lower against the wall but further out along the ground. The angle it makes with the ground is beta.

-Distance between L1 and L2 along the wall equals "b"

-Distance between L1 and L2 along the ground equals "a"

Solution:

H - height of ladder

b - distance ladder slides down wall

a - distance ladder slides along ground (to the right)

*height of alpha ladder - height of beta ladder (from ground) = b

(1) Hsin(alpha) - Hsin(beta) = b      

*distance of beta ladder - distance of alpha ladder (from wall) = a

(2) Hcos(beta) - Hcos(alpha) = a

(1) H[sin(alpha) - sin(beta)] = b

     H = b / [sin(alpha) - sin(beta)]

(2) H[cos(beta) - cos(alpha)] = a

      H = a / [cos(beta) - cos(alpha)]

(1) & (2)

b/[sin(alpha) - sin(beta)] = a/[cos(beta) - cos(alpha)]

a/b = [cos(beta) - cos(alpha)]/[sin(alpha) - sin(beta)]

a/b = -[cos(alpha) - cos(beta)]/-[sin(beta) - sin(alpha)]

a/b = [cos(alpha) - cos(beta)]/[sin(beta) - sin(alpha)]