(a) The accompanying figure shows three triangles. For

Chapter 8, Problem 51

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(a) The accompanying figure shows three triangles. For each triangle, use a calculator to verify that the sum of the cosines of the angles of the triangle is less than 3/2. (b) What is the sum of the cosines of the three angles in an equilateral triangle? (c) Let A, B, and C denote the three angles of a triangle, so that A B C 180. The following sequence of steps proves the inequality cos A cos B cos C 3/2. Supply the reasons or calculations that support each step. [The proof, by W. O. J. Moser, appeared in The American Mathematical Monthly, vol. 67 (1960), p. 695.] (i) cos A cos B cos C 2 cos[(A B)/2] cos[(A B)/2] cos C (ii) 2 cos[(A B)/2] cos C (iii) 2 cos[(180 C)/2] cos C (iv) 2 cos[90 (C/2)] cos C (v) 2 sin(C/2) [1 2 sin2 (C/2)] (vi) (3/2) 2[sin(C/2) (1/2)]2 (vii) 3/2 55 55 70

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