A tower stands on a river bank. From a point on the other bank directly opposite, and at height \(h\) above the water level, the angle of elevation of the top of the tower is \(\alpha\) and the angle of depression of the reflection of the top in the water is \(\beta\). [It is to be assumed that the water is smooth, so that the reflection of any object in the water will appear to be as far below the surface as the tower is above it.] Prove that the height of the top of the tower above the water is \[h\sin(\alpha+\beta)\cosec(\beta-\alpha),\] and the width of the river is \[2h\cos\alpha\cos\beta\cosec(\beta-\alpha).\]

Find these two distances (to the nearest foot) when \(h=\quantity{15}{ft.}\), \(\alpha=24^\circ\), and \(\beta=31^\circ\).

Could we draw a helpful diagram including the reflection of the tower?

Let’s call the height of the tower \(t\), and the width of the river \(w\).

Are there any right-angled triangles that could aid us?

The question says that \(\sin(\alpha+\beta)\) and \(\cosec(\beta-\alpha)\) are in the solution – do we know another way of writing these?