设ABC同时满足sinA+cosB+sinC=0 cosA+cosB+cosC=0的任意角,求证cos^2A+cos^2

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设ABC同时满足sinA+cosB+sinC=0 cosA+cosB+cosC=0的任意角,求证cos^2A+cos^2B+cos^2c=3/2成立

cosa + cosb + cosc = sina + sinb + sinc = 0
(cosa)^2 = (cosb + cosc)^2
= (cosb)^2 + (cosc)^2 + 2*cosb*cosc .(1)
(sina)^2 = (sinb + sinc)^2
= (sinb)^2 + (sinc)^2 + 2*sinb*sinc .(2)
(1) + (2),得cos(b-c) = -1/2
同样可以得到：
cos(c-a) = -1/2
cos(a-b) = -1/2
(1) - (2),得
cos2a = cos2b + cos2c + 2*cos(b+c)
= 2*cos(b+c)*cos(b-c) + 2*cos(b+c) .cos(b-c) = -1/2
= cos(b+c) .(3)
同样可以得到：
cos2b = cos(c+a) .(4)
cos2c = cos(a+b) .(5)
(cosa)^2 + (cosb)^2 + (cosc)^2
= (cos2a + cos2b + cos2c)/2 + 3/2
其中
A = cos2a + cos2b + cos2c
= [(cos2a + cos2b) + (cos2b + cos2c) + (cos2c +cos2a)]/2
= cos(a+b)*cos(a-b) + cos(b+c)*cos(b-c) + cos(c+a)*cos(c-a)
= -[cos(a+b) + cos(b+c) + cos(c+a)]/2
由(3)、(4)、(5)得到
A = cos(a+b) + cos(b+c) + cos(c+a)
所以,A = 0
(cosa)^2 + (cosb)^2 + (cosc)^2 = 3/2
.
附带说明,条件等式 cosa + cosb + cosc = sina + sinb + sinc = 0 描述的是等分圆情况下,求水平坐标(余弦)平方和：将余弦看作是x坐标数值,正弦看作是y坐标.

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