M1410602 - (Section 6.2: The Law of Cosines) 6.09 SECTION...

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(Section 6.2: The Law of Cosines) 6.09 SECTION 6.2: THE LAW OF COSINES PART A: THE SETUP AND THE LAW Remember our example of a conventional setup for a triangle: Observe that Side a “faces” Angle A , b faces B , and c faces C . The Law of Cosines For such a triangle: c 2 = a 2 + b 2 2 ab cos C Think / For Memorization Purposes: This looks like the Pythagorean Theorem, except that there is a third term. “The square of one side equals the sum of the squares of the other two sides, minus twice their product times the cosine of the angle included between them.” Notice that the formula is symmetric in a and b ; we have 2 ab in the formula as opposed to 2 bc or 2 ac . Angle C is the one we take the cosine of, because it is the “special” angle that faces the side indicated on the left. The (very involved) proof is on p.469 of Larson . There is a nicer “Proof Without Words” available on my website.
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(Section 6.2: The Law of Cosines) 6.10 PART B: A THOUGHT EXPERIMENT How does the formula gibe with our geometric intuition? Let’s say we fix lengths a and b , but we allow the other parts to vary. Imagine rotating the side labeled a about the point C so that Angle C changes. If C is a right angle (left figure above), then we obtain the Pythagorean Theorem as a special case: c 2 = a 2 + b 2 2 ab cos π 2 = 0    = 0     c 2 = a 2 + b 2 If C is acute (middle figure above), then: c 2 = a 2 + b 2 2 ab cos C > 0 < 0   c 2 < a 2 + b 2 This reflects the fact that c in this case is smaller than c in the right angle case. If C is obtuse (right figure above), then: c 2 = a 2 + b 2 2 ab cos C < 0 > 0   c 2 > a 2 + b 2 This reflects the fact that c in this case is larger than c in the right angle case.
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(Section 6.2: The Law of Cosines) 6.11 PART C: VARIATIONS OF THE LAW The form given in Part A is the only one you need to memorize, but you should be aware of variations. There is nothing “special” about side c and Angle C . “Role-switching” yields analogous formulas for the other side-angle pairs. Variations of the Law of Cosines a 2 = b 2 + c 2 2 bc cos A b 2 = a 2 + c 2 2 ac cos B Without loss of generality, the proof of one variation yields the others, as well.
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M1410602 - (Section 6.2: The Law of Cosines) 6.09 SECTION...

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