The angular elevation of the summit of a mountain is measured from three points on a straight level road. From a point due south of the summit the elevation is \(\alpha\), from a point due east of it the elevation is \(\beta\), and from the point of the road nearest to the summit the elevation is \(\gamma\).

If the direction of the road makes angle \(\theta\) east of north, prove that

  1. \(\tan\theta=\tan\alpha\cot\beta\)

Before starting, it’s always a good idea to sketch out the problem. Here, we might do that as if we’re looking down on it. We could define \(h\) as the height of the mountain, and then draw three more triangles with the elevation labelled.

Can you use the three different elevation triangles to get three different expressions for the mountain’s height \(h\)?

  1. \(\tan^2\gamma=\tan^2\alpha+\tan^2\beta\).

Are there any useful trigonometrical identities that we can use?