Question

在三角形ABC中,若a^2+b^2=2c^2,求(sinA^2+sinB^2)/sinC^2

Answer 1

根据三角形正弦定理

a/sinA=b/sinB=c/sinC

设a/sinA=b/sinB=c/sinC=1/k

则有sinA=ak

sinB=bk

sinC=ck

(sinA^2+sinB^2)/sinC^2 =(a²k²+b²k²)/c²k²=(a²+b²)/c²

因为a^2+b^2=2c^2

所以(sinA^2+sinB^2)/sinC^2 =(a²k²+b²k²)/c²k²=(a²+b²)/c²=2