sinA+sinB=1/3 cosA+cosB=1/4 求解sin2(A+B)的值

题目

sinA+sinB=1/3 cosA+cosB=1/4 求解sin2(A+B)的值

答案

sinA+sinB=1/3 cosA+cosB=1/4
2sin(a+b)/2 cos(a-b)/2=1/3
2cos(a+b)/2 cos(a-b)/2=1/4
tan(a+b)/2=4/3
tan(a+b)=2tan(a+b)/2 / 1-tan^(a+b)/2
=2*4/3 / (1-(4/3)^2)
=24/7
sin2(a+b)=2tan(a+b)/1+tan^(a+b)=2*24/7 / (1+(24/7)^2)=.