Question

已知cos(a-β)=3/5,sinβ=-5/13,且a∈(0,π/2),β∈

已知cos(a-β)=3/5,sinβ=-5/13,且a∈(0,π/2),β∈(-π/2,0),则sina=

Answer 1

已知cos(a-β)=3/5,sinβ=-5/13,且a∈(0,π/2),β∈(-π/2,0),则sina=
cosβ=12/13
cos(a-β)=cosacosβ+sinasinβ=3/5
cosa*12/13+sina*(-5/13)=3/5
60cosa-25sina=39
两边同时平方
3600cos²a=625sin²a+1521+1950sina
3600(1-sin²a)=625sin²a+1521+1950sina
0=4225sin²a+1950sina-2079
十字相乘
65sina
-33
65sina
63
(65sina-33)(65sina+63)=0
因为
a∈(0,π/2),所以sinA＞0
sina=33/65