A vertical tower standing at the corner of a rectangular field subtends angles of 60^\circ and 45^\circ at the two nearer corners. If \theta is the angle that the tower subtends at the farthest corner, then what is \cot \theta equal to?

Explanation

Let the tower height be h. The distances to the nearer corners (forming the rectangle's sides) are h \cot 60^\circ = h/\sqrt{3} and h \cot 45^\circ = h. The distance d to the farthest corner is the diagonal: d = \sqrt{(h/\sqrt{3})^2 + h^2} = h\sqrt{4/3}. Thus, \cot \theta = \frac{d}{h} = \frac{2}{\sqrt{3}}.

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