13问答网 > α属于(π⼀4,3π⼀4)β属于(0,π⼀4),cos(π⼀4 -α)=3⼀5 sin(5π⼀4 +β)=-12⼀13, 求cos(α+β)

α属于(π⼀4,3π⼀4)β属于(0,π⼀4),cos(π⼀4 -α)=3⼀5 sin(5π⼀4 +β)=-12⼀13, 求cos(α+β)

2025-02-23 22:50:43

推荐回答（1个）

回答1：

cos(π/4 - α) = 3/5
sin(π/4 - α) = ±√[1 - (cos(π/4 - α))^2] = ±4/5
α∈(π/4,3π/4)
sin(π/4 - α) = -4/5

sin(5π/4 + β) = -12/13
cos(5π/4 + β) = ±√[1 - (sin(5π/4 + β))^2] = ±5/13
β∈(0,π/4)
cos(5π/4 + β) = -5/13

cos(π/4 - α)cos(5π/4 + β) + sin(π/4 - α)sin(5π/4 + β)
= cos[(5π/4 + β) - (π/4 - α)]
= cos[π + (α+β)]
= -cos(α+β)

cos(π/4 - α)cos(5π/4 + β) + sin(π/4 - α)sin(5π/4 + β)
= 3/5 * (-5/13) + (-4/5) * (-12/13)
= -3/13 + 48 / 65
= 33 / 65

cos(α+β) = - 33 / 65