Question

已知sin^2 A+sin^2 B+1/4=sinAsinB+1/2(sinA+sinB),求锐角A,B的值。

Answer 1

sin^2 A+sin^2 B+1/4=sinAsinB+1/2(sinA+sinB),
4sin^2 A+4sin^2 B+1-4sinAsinB-2sinA-2sinB=0
4[sin^2A-2sinAsinB+sin^2 B]+[4sinAsinB-2sinA-2sinB+1]=0
4[sinA-sinB]^2+(2sinA-1)(2sinB-1)=0
sinA-sinB=0
(2sinA-1)(2sinB-1)=0
sinA=1/2
sinB=1/2
A=B=30°

Answer 2

A=B=30度

Answer 3

(sinA+sinB)^2-1/2(sinA+sinB)+1/4=3sinAsinB
A=B时，(sinB)^2-sinB+1/4=0
(sinB-1/2)^2=0
B=30
A=30
讨论：
sinA=ksinB时 [(k+1)^2-3k](sinB)^2-(k+1)sinB/2+1/4=0
(k^2-k+1)[sinB-(k+1)/(4k^2-4k+4)]^2=(k+1)^2/(16k^2-16k+16)-1/4
(k+1)^2/(4k^2-4k+4)-1/4=(k^2+2k+1)/(16k^2-16k+4)-1/4
4k^2+8k+4>16k^2-16k+4
12k^2-24k<0
k^2-2k<0
0<k<2时,sinA\sinB有解