Measuring by Measuring by TrigonometryTrigonometry

ObjectivesObjectivesWe have to measure the height of

the “Monumento a las Batallas”.

Step by Step.Step by Step.The three boys of the group and

Salva have gone to the “Plaza de las Batallas” by bus. When we got there, we use our AMS to measure the angles in the first position, then we choose another position, we measured the distance between it and the first position and we measured the second angle. Then, we took the bus again and we went to Carlos’ house, where we calculate the height of the monument and then we made this PowerPoint.

MethodsMethodsTo measure the monument, we have

used the trigonometry. Using the AMS (Angle Measuring System) we can obtain the angles from our position to the top of the monument and using a tape measure to obtain the distance between the 2 places where we have measured the angles.

Instruments that we usedInstruments that we usedThe 4 AMSs( Angle Measuring

System, name given to them by Alberto) that are the quarter of a circle with the angles marked, a straw to look and a thread with a coin, button, etc that moves and stays over the angle.

A 50 metres tape measure.Notebook and pen to take notes.Computer.Calculator.

TrigonometryTrigonometry Trigonometry is a branch of Mathematics that studies the relationships between the angles and sides of a triangle. It is used for surveying, navigation (either land, or air, or sea), land measure, etc. At first the Egyptians and Babilonians used it, but they did not know what they were exactly using. Later the Greeks started to use Trigonometry and made it into an ordered science. They used it for Astronomy By the 10th century, Islamic mathematicians were using all six trigonometric functions, they had tabulated their values, and they were applying them to problems in spherical trigonometry. At the same time, Chinese mathematicians developed trigonometry independently. Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew towards a major branch of Mathematics.

TrigonometryTrigonometry

In trigonometry we use sines, cosines and tangents to relate angles and sides. Their ecuations are:

Sine= Opposite/HypotenuseCosine= Adjacent/HypotenuseTangent= Opposite/ Adjacent

With these operations, if we know an angle and a side we can obtain the other sides. If we have two sides, by the inverse of the sine, cosine or tangent, we can obtain the different angles.

Process we have followedProcess we have followed

We can do more things with trigonometry. Now we are going to explain the process that we have done.To measure the height of the statue, we have measured two angles. These angles are measured with the AMS. We have also measured the distance between the two angles.With these measures we do an equation system.

Tan β = h/dTan α = h/ d+ d’

We have used the tangent because this relates the opposite, which is the height of the statue, and the adjacent, which is the floor,that can be measured.

MeasurementMeasurement With four different AMSs, we have measured the angles

and then we calculate the average angle.

The distance between the two angles (d’) is 11.95 m.

CalculationsCalculationsTan 60° = h/dTan 39°15’ = h/d + 11’95d · Tan 60 ° = (d + 11.95) · Tan 39 °15’d(Tan 60 ° - Tan 39 °15’) = 11.95 · Tan

39°15’d = (11.95 · Tan 39°15’) / (Tan 60° -

Tan 39 °15’)d = 10.67h = 10.67 · Tan 60h = 18.48

ErrorsErrorsDuring the process we have commited

some errors due to the accuracy of the AMS.

Alpha errors:Absolute error is + 1°.Relative error is 1.666...%Beta errors:Absolute error is + 1°15’.Relative error is 3.1847%.Relative error = (absolute error / extract value) · 100%

Absolute error = +(avarage – furthest value)

Summing-upSumming-upTrigonometry is an useful method to

calculate the height of buildings, mountains, etc. This is also useful for triangulation in GPS.

We think that our reasoning is good, but we also believe that the AMS has commited some errors.

The reason why it has been used such as for a long time says us that this is accurate and useful.

FINFINProject made by:-Alberto Marín Caba-Carlos Ruiz Soriano-Salvador López de Moral

-Javier Hermoso Romero