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    • Thread Starter

    Q: The points A,B & C lie in a straight line with AB=x & BC=2x.
    Avertical tower OH is of height h and its base Olies in the same
    horizontal plane as ABC but not on the line ABC.
    The angle of elevation of H from A,B & C are alpha, beta & alpha

    (i) prove thah h^2(cot^2(alpha) - cot^2(beta)) = 2x^2
    (i) I start by taking the triangle AOB where AB = x and AHO & BHO
    (ii)Because I'm required to prove using cot I'm going to use tan.
    tan (alpha) = h/AO
    AO = h/tan (alpha)
    AO = h*cot (alpha)

    tan (beta) = h/BO
    BO = h/tan (beta)
    BO = h*cot (beta)

    tan (alpha) = h/CO
    CO = h/tan (alpha)
    CO = h*cot (alpha)

    working out that in triangle CBO the angle is unkown called theta and triangle ABO the angle must be 180 - theta

    (iii) using cosine law with Cos (theta as the subject)

    Cos (theta) = {(2x)^2 + (BO)^2 - (CO)^2}/ 2*(2x)*(BO)


    Cos (180 -(theta)) = {(x)^2 + (BO)^2 - (AO)^2}/ 2*(x)*(BO)

    Therefore, Cos (180 -(theta)) = -cos (theta)

    therefore {(O) - (BO)^2 - (x)^2}/ 2*(x)*(BO)}

    Replacing AO,BO & CO for the cot values these two equations shouldbe equal to one another. I don't seem to get the right answer after manipulating them though

    Can you show me how this is proven? There are other parts to the
    question but I think once I understand this I will be okay on my own.

    many thanks
    • Study Helper

    Study Helper
    (Original post by dojos)
    Cos (theta) = {(2x)^2 + (BO)^2 - (CO)^2}/ 2*(2x)*(BO)

    Cos (theta) = {(AO)^2 - (BO)^2 - (x)^2}/ 2*(x)*(BO)}
    Assuming I've corrected the odd typo and what you've got is as in the "quote" above, then it does work out, and you've made a slip in the susequent manipulation somewhere.

    First step gives me:

    {(2x)^2 + (BO)^2 - (CO)^2}/ 2 = (AO)^2 - (BO)^2 - (x)^2


    4x^2 + (BO)^2 - (CO)^2 = 2(AO)^2 - 2(BO)^2 - 2(x)^2


    6x^2 = 2(AO)^2 +(CO)^2 - 3(BO)^2


    6x^2 = 2 h^2*cot^2 (alpha) +h^2*cot^2 (alpha) - 3h^2*cot^2 (beta)

    • Thread Starter

    Thanks so much Ghostwalker - I just couldn't see the wood through the trees! I think I'll practise my algebra.
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