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Unformatted text preview: a,b,c > 0 and angle γ between legs a and b , see the ﬁgure below. We want to prove the law of cosines c 2 = a 2 + b 2-2 ab cos γ , (1) and derive the triangle inequality. (i) Consider the vectors u = ± a ² and v = b ± cos γ sin γ ² . Show that k u k = a and k v k = b . (ii) Plot the vectors u and v and show that they span the triangle with sides a,b,c . Conclude that c = k u-v k and use this to derive (1). (iii) Use the law of cosines to derive the triangle inequality in the form c 2 ≤ ( a + b ) 2 , and determine for which γ ∈ [0 ,π ] we have equality. a b c γ 2...
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