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f.4 maths (M2數)....

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發問:

1. Prove that cos(x+y)cos(x-y) = cos^x+cos^y-1. 2. In triangle ABC, it is given that 2sinAcosB = sinC. Prove that triangle ABC is an isos. triangle.

最佳解答:

1) Applying product to sum formula: cos (x + y) cos (x - y) = (1/2) {cos [(x + y) + (x - y)] + cos [(x + y) - (x - y)]} = (1/2) [cos 2x + cos 2y] = (1/2) [(2 cos2 x - 1) + (2 cos2 y - 1)] = (1/2) (2 cos2 x + 2 cos2 y - 2) = cos2 x + cos2 y - 1 2) 2 sin A cos B = sin C sin (A + B) + sin (A - B) = sin C sin (180 - C) + sin (A - B) = sin C sin C + sin (A - B) = sin C sin (A - B) = 0 A - B = 0 since 0 < A, B < 180 A = B Hence △ABC is an isos.△.

其他解答:

2sinAcosB=sinC sinAcosB=sin(C/2)cos(C/2) ==> B<90 then A=C/2=B|||||2sinAcosB = sinC A = B