G5 trigonometry


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G5 trigonometry

  1. 1. OUR JOURNEYTO THETRIGONOMETRY By: Gema Cabrera Gissella Garriga Elvira Marín Laura Olmo
  2. 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. 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. 4. HISTORYPtolemy provided what is now known as Menelauss 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. 5. HISTORYThe 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. 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. 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. 8. Names 1st measurement 2nd measurementElvira 37º 26ºGema 37º 26ºGissella 36º 27ºLaura 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. 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. 10. CONCLUSIONWe 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.