Trigonometric function

用戶154568

2007-05-14 1:53 am

In triangle ABC, given that
(sin A)^2-(sin B)^2-(cos C)^2=2sin A cos B sin C -1
if cos B=sin A sin C
prove ABC is a right-angled triangle

回答 (1)

2007-05-14 5:24 am

✔ 最佳答案

if cosB = sinAsinC,

sin²A - sin²B - cos²C =2sinAcosBsinC - 1
sin²A - (1- cos²B) - (1 - sin²C) = 2sinA(sinAsinC)sinC - 1
sin²A + cos²B + sin²C = 2sin²Asin²C + 1
sin²A + (sinAsinC)² + sin²C = 2sin²Asin²C + 1
sin²A + sin²Asin²C + sin²C = 2sin²Asin²C + 1
sin²A - sin²Asin²C - 1 + sin²C = 0
sin²A(1 - sin²C) - (1-sin²C) = 0
(1-sin²C)(sin²A - 1) = 0
1 - sin²C = 0　or　sin²A - 1 = 0
sin²C = 1　or　sin²A = 1
sinC = ± 1　or　sinA = ± 1

As ABC is a triangle, each angle in triangle ABC < 180°
C = 90°or　A = 90°

Therefore, ABC is a right-angled triangle.

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