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The Law of Cosines: A Generalization of the Pythagorean Theorem
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The Pythagorean Theorem provides a method to find a missing side for a right triangle. But what do we do for triangles that are not right? The law of cosines states that: c2 = a2 + b2 - 2ab cosγ. Notice that if γ = 90°, the equation reduces to the Pythagorean Theorem since cos90° = 0. Why do we use the Law of Cosines? Figure 1
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The law of cosines can be useful in triangulation, a process for solving a triangles unknown sides and angles when only certain information about that triangle is given. For example, it can be used to find: the third side of a triangle if two sides are known and the angle between them is also known. the angles of the triangle if one knows the three sides. the third side of a triangle if two sides are known and an opposite angle to one of those sides is known. Applications
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The cosine function first arose from the need to compute the sine of the complementary angle (90° - α). The general law of cosines was not developed until the early part of the 10th century by Muslim mathematicians. Brief History of the Law of Cosines Figure 2
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An early geometric theorem which is equivalent to the law of cosines was written in the 3rd century B.C. in Euclid’s Elements. This theorem stated the following: Proposition 12In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. — Euclid's Elements, translation by Thomas L. Heath. Using the figure below, Euclid’s statement can be put into the following algebraic terms: AB2 = CA2 + CB2 +2(CA)(CH) Law of Cosines in Euclid’s Elements Figure 3
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There are many proofs for the Law of Cosines that use: The distance formula Trigonometry The Pythagorean Theorem Ptolemy’s Theorem Area Comparison Circle Geometry Vectors Proofs for the Law of Cosines
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To demonstrate your knowledge of the Law of Cosines and how it is a direct result of the Pythagorean Theorem, please answer one of the questions in the assessment section titled “3a - Assessment” which can be found in the sidebar on the left of this page. Learning Activities
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