Question

在△ABC中,其三边a,b,c满足:c^4-2(a^2+b^2)c^2+a^4+b^4-a^2b^2=0,则角C的大小为

Answer 1

c^4-2(a^2+b^2)c^2+a^4+b^4-a^2b^2=0

c^4-2(a^2+b^2)c^2+a^4+b^4+2a^2b^2-3a^2b^2=0

c^4-2(a^2+b^2)c^2+(a^2+b^2)^2-3a^2b^2=0

(c^2-a^2-b^2)^2-(√3ab)^2=0

(c^2-a^2-b^2-√3ab)(c^2-a^2-b^2+√3ab)=0

c^2-a^2-b^2=√3ab or -√3ab

由余弦定理

cosC=(a^2+b^2-c^2)/2ab=±√3/2

所以C=π/6 or 5π/6