已知sin(3π-α)=根号2倍的sinβ,根号3倍的cos（-α）=-根号2倍的cos(π+β）,0＜α＜π

题目

已知sin(3π-α)=根号2倍的sinβ,根号3倍的cos（-α）=-根号2倍的cos(π+β）,0＜α＜π
0＜β＜π,求α、β的值

答案

sin(3pi-a) = sin(pi-a) = sin(a)
=> sin(a) = 根号2 * sin(b)
根号3 * cos(-a) = 根号3 * cos(a)
-根号2 * cos(pi+b) = 根号2 * cos(b)
=> cos(a) = 根号（2/3）* cos(b)
因为 sin(a)^2 + cos(a)^2 = 1
所以,2sin(b)^2 + 2/3 * cos(b)^2 = 1
=> 2sin(b)^2 + 2/3 * [1-sin(b)^2] = 1
=> 4/3 sin(b)^2 = 1/3
sin(b)^2 = 1/4
sin(b) = +/- 1/2
因为 0 sin(b)=1/2 => b=pi/6 或者 5pi/6
sin(a) = 根号2 * sin(b) = (根号2)/2
=> a = pi/4 或者 3pi/4