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. \(\tan\theta=\tan\alpha\cot\beta\)
The road with the points A (due south of the mountain), B (due east), C (closest), and O (the mountain). lengths are marked as follows: AO, 1; CO, sin theta; BO, tan theta. Angle OAC = angle COB = theta.
Three right-angled triangles, AOM, BOM and COM, with bases AO, BO and CO respectively, and heights all h, and angles given by the angles of elevation (so e.g. angle OAM = alpha).

Let us choose the length \(AO\) to be \(1\), without loss of generality. (This says effectively that our angles will be the same whatever \(AO\) is.)

Let the height of the mountain be \(h\). We have three expressions for the height \(h\): \[\begin{align*} \triangle AOM:\quad h&=\tan\alpha\\ \triangle BOM:\quad h&=\tan\theta\tan\beta\\ \triangle COM:\quad h&=\sin\theta\tan\gamma. \end{align*}\]

Equating \(h\) from triangles \(AOM\) and \(BOM\), we get \[h=\tan\alpha=\tan\theta\tan\beta\] so \[\tan\theta=\tan\alpha\cot\beta\] as required.

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

We will use here the identity \[1+\cot^2\theta=\cosec^2\theta.\]

We have from the diagrams

  1. \(\tan^2 \gamma = \dfrac{h^2}{OC^2}\)
  2. \(\tan^2\alpha = h^2\)
  3. \(\tan^2\beta = \dfrac{h^2}{OB^2},\) so
\[\begin{align*} \tan^2\alpha + \tan^2\beta &= h^2 + \dfrac{h^2}{OB^2} \\ &=h^2\left(1 + \dfrac{1}{OB^2}\right)\\ &= h^2(1+\cot^2\theta)\\ &= h^2\cosec^2\theta\\ &= \dfrac{h^2}{\sin^2\theta}\\ &= \tan^2\gamma. \end{align*}\]

Find \(\gamma\), if \(\theta=31^{\circ}\) and \(\alpha=8^{\circ}\).

We can eliminate \(\beta\) from the results of (i) and (ii) to get \[\tan^2\gamma=\frac{\tan^2\alpha}{\sin^2\theta}.\] Substituting in the values for \(\theta\) and \(\alpha\), we find \(\gamma\approx15.3^\circ\).