The document discusses the history of trigonometry and the group's project to use trigonometry to measure the height of a church. It provides background on important historical figures who developed trigonometry like Hipparchus, Ptolemy, and Muslim astronomers. It then describes how the group used a homemade quadrant and trigonometric calculations to determine the church was approximately 14.57 meters tall. The group felt the project was a good learning experience but took a long time to complete carefully.

1. OUR JOURNEY
TO THE
TRIGONOMETRY
By: Gema Cabrera
Gissella Garriga
Elvira Marín
Laura Olmo

2. INTRODUCTION
 A “brief” text about the history of trigonometry,
with some pictures to show the examples.
 Materials.
 Calculate of the average, absolute errors and the
relative errors.
 The process of finding the distance between the
church and us, and then, the height of the
church.
 Conclusion.

3. HISTORY
 The history of trigonometry goes back to the earliest recorded
mathematics in Egypt and Babylon. The Babylonians established the
measurement of angles in degrees, minutes, and seconds. Not until the
time of the Greeks, however, did any considerable amount of trigonometry
exist. In the 2nd century BC the astronomer Hipparchus compiled a
trigonometric table for solving triangles. Starting with 71° and going up to
180° by steps of 71°, the table gave for each angle the length of the chord
subtending that angle in a circle of a fixed radius . Such a table is
equivalent to a sine table.
In his great astronomical handbook, The Almagest, Ptolemy provided a
table of chords for steps of 1°, from 0° to 180°, that is accurate to 1/3600 of
a unit. He also explained his method for constructing his table of chords,
and in the course of the book he gave many examples of how to use the
table to find unknown parts of triangles from known parts.
AC BD = AB CD + AD BC

4. HISTORY
Ptolemy provided what is now known as Menelaus's theorem for solving
spherical triangles, as well, and for several centuries his trigonometry was
the primary introduction to the subject for any astronomer.
At perhaps the same time as Ptolemy, however, Indian astronomers had
developed a trigonometric system based on the sine function rather than
the chord function of the Greeks. This sine function, unlike the modern
one, was not a ratio but simply the length of the side opposite the angle in
a right triangle of fixed hypotenuse. The Indians used various values for
the hypotenuse. Late in the 8th century, Muslim astronomers inherited
both the Greek and the Indian traditions, but they seem to have preferred
the sine function. By the end of the 10th century they had completed the
sine and the five other functions and had discovered and proved several
basic theorems of trigonometry for both plane and spherical triangles.
Sin CE/ Sin EA= Sin CF/ Sin FD Sin BD/Sin BA
Sin CA/ Sin EA= Sin CD/ Sin FD Sin BF/Sin BE

5. HISTORY
The Muslims also introduced the polar triangle for spherical triangles. All of
these discoveries were applied both for astronomical purposes and as an
aid in astronomical time-keeping and in finding the direction of Mecca for
the five daily prayers required by Muslim law. Muslim scientists also
produced tables of great precision.
Finally, in the 18th century the Swiss mathematician Leonhard Euler
defined the trigonometric functions in terms of complex numbers (see
Number). This made the whole subject of trigonometry just one of the
many applications of complex numbers, and showed that the basic laws of
trigonometry were simply consequences of the arithmetic of these
numbers.

6. THE BEGINNING OF THE PRACTISE
 We want to measure the height of Salvador’s
church (Barcelona Avenue).
 Materials that we have used:
-Tape measure.
- Quadrant (cuadrante):
* Cardboard.
* Graph paper.
* Thread.
* Buttons.
* Straw.
- Camera.

7. LOOKING FOR…..THE HEIGHT
 We stayed in front of the church. First, all of us
look through the quadrant from our eyes to the
highest point of the cross of Salvador’s church.
We didn’t measure the distance between the
church and the measurement of the angles.
From that point we added 10 meters more and we
looked through the quadrant again.

8. Names 1st measurement 2nd measurement
Elvira 37º 26º
Gema 37º 26º
Gissella 36º 27º
Laura 35º 25º
- FIRST:
THE AVERAGE: 36º+37º+37º+35º=145º;145º:4=36.25
36.25=36º15’
ABSOLUTE ERROR: 1.25
RELATIVE ERROR: 3.45
- THE SECOND:
THE AVERAGE: 26º + 27º + 26º + 25º= 104º: 4 = 26º
ABSOLUTE ERROR: 2
RELATIVE ERROR: 7’69

9. THE PROCESS
 We want to know the opposite side of the angle
36º15’ and we don’t know its adjacent. We have,
more or less, the adjacent of the second angle
which is 26º. So, we measure their tangent:
tan 36º15’= h/x h=tan36º15’ · x
tan 26º = h/x+10 h= tan26º · (x + 10)
tan 36º 15’· x = tan 26º (x + 10)
x = 10 · tan 26º/ tan 36º15’-tan 26º
x = 19.87 m
h= tan 36º15’ · 19.87= 14’57 m

10. CONCLUSION
We think that it was a good idea to divide the
project in different parts. It was important for us
to write about the history of trigonometry but
there was too much information so we had to
make the summery.
It took a long time and it was a bit difficult for us
because we had to do it slowly and to think the
correct and exact information, pictures and
answers.
We believe that it has good points such as that we
are going to learn and remember the concepts of
this unit better than if we had done an exam.