Engineering Calculus Notes 32

Engineering Calculus Notes 32 - In acute-angled triangles...

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20 CHAPTER 1. COORDINATES AND VECTORS (d) Now combine these equations with Equation ( 1.12 ) to prove Pythagoras’ Theorem. The basic proportions here are those that appear in Euclid’s proof of Proposition 47, Book I of the Elements , although he arrives at these via diFerent reasoning. However, in Book VI, Proposition 31 , Euclid presents a generalization of this theorem: draw any polygon using the hypotenuse as one side; then draw similar polygons using the legs of the triangle; Proposition 31 asserts that the sum of the areas of the two polygons on the legs equals that of the polygon on the hypotenuse. Euclid’s proof of this proposition is essentially the argument given above. 16. The Law of Cosines for an acute angle is essentially given by Proposition 13 in Book II of Euclid’s Elements [ 27 , vol. 1, p. 406] :
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Unformatted text preview: In acute-angled triangles the square on the side subtending the acute angle is less than the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut oF within by the perpendicular towards the acute angle. Translated into algebraic language (see igure 1.15 , where the acute angle is ABC ) this says A B C D igure 1.15: Euclid Book II, Proposition 13 | AC | 2 = | CB | 2 + | BA | 2 | CB || BD | . Explain why this is the same as the Law of Cosines. 1.2 Vectors and Their Arithmetic Many quantities occurring in physics have a magnitude and a directionfor example, forces, velocities, and accelerations. As a...
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