1. RY
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2. WHAT IS TRIGONOMETRY?
Trigonometry is a part of elementary
mathematics that studies angles, triangles
and trigonometric functions such as sine
(abbreviated sin), cosine (abbreviated
cos) and tangent (abbreviated tg).

3. SIN
The sin of an angle is the
relation between its opposite
side and the hypotenuse. The
sin is directly proportional to
the opposite side and
inversely proportional to the
hypotenuse, so we can guess
it by the following ecuation:
Sin = opposite/hypotenuse

4. COSINE
The cosine of an angle is the relation
between its adjacent side and the
hypotenuse. The cosine is directly
proportional to the adjacent side and
inversely proportional to the
hypotenuse, so we can guess it by the
following ecuation:
Cosine = adjacent/hypotenuse

5. TANGENT
The tangent of an angle is the
relation between its opposite side
and its adjacent side. The tangent
is directly proportional to the
opposite side and inversely
proportional to the adjacent side,
so we can guess it by the following
ecuation:
Tangent = opposite/adjacent

6. INVERSAL FUNCTIONS
All this functions (sin, cosine and tangent),
have their own inversal functions, which
let us know the angle which they come
from. For example, if the cosine of 20º is
0.93969262, the inversal cosine (cos^-1)
of 0.93969262 is 20º.

7. TRIGONOMETRY WITH CALCULATOR
For knowing the values of the different trigonometric
functions of an angle with the help of a calculator, we
have to push the button of the function we want to know
(sin for sin, cos for cosine and tan for tangent) and then
write the degrees of the angle we’re working with. For the
inversal functions, we have to push “shift” and the button
of the function of we we want to know its inverse.
Sometimes we need some of the results are given in
degrees, so for it we have to write the number which
value we want to know in degrees and push the º’’’
button.

8. USE OF TRIGONOMETRY
Trigonometry is used for measuring
some heights from which we can
only know some of the angles
between it and the floor, or to know
distances between some places
knowing some angles between them.
In the past it was also used for sea
orientation, but know is not longer
used.

9. HISTORY OF TRIGONOMETRY
The trigonometry was first used by Egyptians and Babylonians for agriculture,
building pyramids and astronomy. The eqyptians established the measure of
angles in grades, minutes and seconds. Then Greeks used it mainly in
astronomy, and their concepts were lately used by the Arabians, who in the
VIII century improved it with new teories and functions. Some centuries later,
trigonometry improved with the discorverment of the logarithms, by John
Napier, and other important discoverments by Newton and Leonhard Euler.

10. RY
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R IG É
T JA

11. MEASURING STEPS
We have gone to the cathedral to measure the high of it.
Now we are going to explain all the steps that we have follow:
1.We arrived to its back- square, because the main- square (Plaza de
Santa María) was under construction, so we couldn’t stay there.
2. We stopped in the pavement . The square was opposite us. In the
middle there was a road.
3. Then, we took the measure machine and looked throught the straw
to the highest part of the cathedral (the highest part we reached see).
4. From that position (in the pavement behind the road) it was 45º.
5. After, we crossed the road and it was 50º.
6. Later, Alberto crossed the street again by the measure of his steps, it
was o,77 each one the total was 8.47m (11 steps).

12. PLAIN OF CATHEDRAL BACK-SQUARE

13. THE PROCESS OF CALCULATION
Tg 50º= h/X
Tg 45º= h/8.47+X
Tg 50º X = h
Tg45º = Tg 50 X/8.47 + X
Tg 50º = h/44.1720 x tg 45º (8,47+x) =tg 50º X
Tg 45º (8,47+ X) = Tg 50º X
Tg 45º x 8.47 + Tg 45º x X = Tg 50º X
(Tg 45º = 1)
8,47 + X = Tg 50 º X
8,47 = Tg 50 º - X
8,47 = 0,19175 x X
8,47/0,19175 = X
44,172 = X
Tg 50º = h/44,172
Tg 50º x 44,172 = h
h= 52, 6421

14. HOW WE DID IT?
We made a system of ecuations, with the height of the cathedral (h)
and the length of the back square (X) as the unknown numbers. If
we know the measure of the angles (50º and 45º) we have to
calculate their tangents with the calculator and use them to solve
the system. However, at the end we realised that we were taking
that measure from the height of our eyes, so we added 1.55 meters
to the final result, which is the distance between Dani’s eyes and
feets, because he was the person who was measuring. So at the
end, the cathedral’s back height is of 54.1921 meters.

15. OUR OWN CONCLUSION
We think that this project has helped us to
understand what is trigonometry used for,
because we have had the opunity of using it
in a real situation, and we also think that it
has helped us to improve our trigonometry
knowledges, because I think you can learn
things better in a funny way. And, of course,
we have had a really good time together,
and we hope this project will help us to get a
good maths mark and to make trigonometry
funnier.

16. N D
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17. CT
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18. N
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