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Pythagoras

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Pythagoras theorem

Abstract
Pythagoras theorem gives a relationship of the three sides of right angled triangles. It is extended to draw relationship among the interior angles of such right-angles triangles to form what is known as trigonometrical ratios. The theorem has vast application in science and mathematical phenomena. It is also used in the derivation of other theorems. This paper attempts to uniquely explain the theorem by experiment. Calculations and measurements will be done to arrive at stated proofs. I addition, theoretical values (value obtained through calculation) and practical ones are compared to establish the degree of error so allowed.
Introduction
Pythagoras theorem is mathematically expressed as So that c is the square root of the first two terms

The sides as labeled are: a is the adjacent, b the opposite and c the hypotenuse.
Therefore, the square of the hypotenuse is equal to the sum of the squares of the opposite and the adjacent. The adjacent can be called the base and the opposite the height of the triangle. These two sides are often referred to as the legs of the triangle and the hypotenuse as the longest side of the triangle.
Relationships of the interior angles
This is basically the trigonometrical ratios. Included angle is the angle enveloped by any two sides in the triangle. We use capital letters to denote the angles so that A is the angle included by band c. Similarly the rest will be B and C. stated otherwise, c corresponds to angle C and b corresponds to angle B.
Therefore, the ratios and their names are as follows:
Sine A = bc; opposite hypotenuse
Cosine A= ac; adjacenthypotenuse
Tangent A= ba; oppositeadjacent
The ratios for the other angles (Band C) will be obtained in a similar manner.
From the above ratios, angles A, B and C can be obtained by getting the inverse of the ratios. For instance;
A = sine-1 ( bc )
Experiment
Five right angled triangles were drawn in a piece of graph paper. Such were different in sizes to reflect different side lengths and different two base angles since the third must always be a right angle. After drawing, a ruler was used to measure the lengths o the sides. Further, a protractor was used to measure the values of the two non right angles in the triangles.
Calculated and measured values were compared to establish the degree of error so allowed.
The table below gives the results thus obtained. Triangle | Measurant | Calculated | Measured | R | 1 | Ѳ | 19.660 | 200 | 1.75% | | Φ | 69.980 | 69.50 | 0.7% | | hypotenuse | 7.43cm | 7.45cm | 0.3% | 2 | Ѳ | 53.50 | 53.10 | 0.75% | | Φ | 400 | 36.90 | 8.4% | | hypotenuse | 2.5 cm | 2.5 cm | 0.0% | 3 | Ѳ | 18.20 | 220 | 17% | | Φ | 6.60 | 700 | 0.6% | | hypotenuse | 3.2 cm | 3.2 cm | 0.0% | 4 | Ѳ | 75.40 | 680 | 10.9% | | Φ | 18.80 | 250 | 24.9% | | hypotenuse | 3.2 cm | 3.1 cm | 3.2% | 5 | Ѳ | 38.70 | 400 | 3.25% | | Φ | 51.40 | 600 | 14.3% | | hypotenuse | 4.6 cm | 4 cm | 15% |

The values of the third column were calculated as follows:
Ѳ = tan-1 ( oppositeadjacent)
Φ=sin-1 (opposite hypotenuse)
The hypotenuse was obtained by using the relation
Hypotenuse= adjacent2+opposite2
The value R in the 5th column is used to give percentage error; the percentage by which the measured and the calculated values differ.
It was obtained by getting (1- calculatedmeasured) x 100%
Precautions taken during measurements
The error of parallax was avoided by ensuring that ruler and protractor were placed so that the eye was directly above the markings. Lines for the triangles were drawn using sharp pointed pencil leads. Thinness ensured of the lines so produced would minimize error in the measurements.
Conclusions
Pythagorean Theorem was validated by help of an experiment. Trigonometrical ratios were also established in which values were calculating d and measured. This was done for different triangles and values so obtained were compared to establish the deviation between the two. Pythagoras’ theorem thus holds.…...

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