A tower subtends an angle α at a point P on the same level as the foot of the tower. Q is a point vertically above P and PQ = h. If the angle of depression of the foot of the tower measured from Q is β, then what is the height of the tower?
h \(\tan\alpha\cot\beta\)
Let A be the foot and B the top of the tower, with \(AP=d\) and tower height \(H=AB\). Since Q is at height h above P and the angle of depression of A from Q is β, \(\tan\beta=\dfrac{h}{d} \Rightarrow d = h\cot\beta\). Since the tower subtends angle α at P (P being at ground level), \(\tan\alpha = \dfrac{H}{d}\), so \(H = d\tan\alpha = h\cot\beta\tan\alpha = h\tan\alpha\cot\beta\).
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