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).
SINThe 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
COSINEThe 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
TANGENTThe 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
INVERSAL FUNCTIONSAll 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.
TRIGONOMETRY WITH CALCULATORFor 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.
USE OF TRIGONOMETRYTrigonometry 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.
HISTORY OF TRIGONOMETRYThe 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.
MEASURING STEPSWe 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).
THE PROCESS OF CALCULATION Tg 50º= h/X Tg 45º= h/8.47+X Tg 50º X = h Tg45º = Tg 50 X/8.47 + XTg 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
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.
OUR OWN CONCLUSIONWe 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.