# Trigonometry (non right angled triangles)

The height of a vertical tower is to be found by a surveyor. The angle of elevation of the top of the tower from a point on the horizontal ground some distance away is measured as 28.7 degrees. From this point the surveyor moves 21.6 metres directly towards the tower and repeats the measurement. The angle of elevation of the top of the tower from this point is 68.4 degrees. Find the height of the tower.

Can I use SINE rule for this? If so can someone please point out which angles/sides will be what figures. ie.is bc=21.6cm? etc Thankyou The answer given is 15.098cm.

Let $x$ be the distance towards the tower during the first measurement. Let $y$ be the height of the tower.

The tangent ratio during the first measurement is $$\tan(28.7^\circ)=\dfrac{y}{x}$$

During the second measurement, the distance towards the tower is $x-21.6$. The tangent ratio changed to $$\tan(68.4^\circ)=\frac{y}{x-21.6}$$

This gives two equations with two unknowns, from which we want to obtain $y$.

From the first equation, we get $x=\dfrac{y}{\tan(28.7^\circ)}$. Substituting into the second equation, we get $\tan(68.4^\circ)=\dfrac{y}{\frac{y}{\tan(28.7^\circ)}-21.6}$.

This gives $y\approx15.1$. Therefore, the length of the tower is approximately $15.1$ meters.

• Thankyou but i forgot to add that the answer given is 15.098cm, which is what confused me as i got 32.456cm. So I am unsure how to do this? – Amy Oct 26 '14 at 19:35
• The measurement in cm seems odd when talking about towers.. Are you sure you have the right answer? @Amy – rae306 Oct 26 '14 at 19:37
• Sorry i meant meters. Yes it was a computer generated question and answer. – Amy Oct 26 '14 at 19:38
• @Amy, sorry I had my calculator on radian mode.. The correct answer is indeed $15.1$ meters. Can you tell me where you are stuck in my answer? – rae306 Oct 26 '14 at 19:42
• Thankkyou, and instead of treating it as x-26.1 I used 21.6 which is where i went wrong. Thankyou for your time :) – Amy Oct 26 '14 at 19:43