化简化简：1.1／2COSx－二分之根号三sinx 2.根号三sinx+cosx 3.根号二（sinx-c第三道题是根号二(sinx-cosx)4.根号二cosx-根号六sinx5.已知sin(α-β）cosα-cos(β-α)sinα=3／5,β是第三象限角，求sin（β+5π／4）

Question

化简化简：1.1／2COSx－二分之根号三sinx 2.根号三sinx+cosx 3.根号二（sinx-c
第三道题是根号二(sinx-cosx)
4.根号二cosx-根号六sinx
5.已知sin(α-β）cosα-cos(β-α)sinα=3／5,β是第三象限角，求sin（β+5π／4）的值

Answer 1

1.
(cosx)/2－(√3sinx )/2
=cosx·cos(π/3)-sinx·sin(π/3)
=cos(x+π/3)=cos(x+60°)
(cosx)/2－(√3sinx )/2
=cosx·sin(π/6)-sinx·cos(π/6)
=sin(π/6-x)=sin(30°-x)
2.
Asinα+Bcosα=√(A²+B²)sin(α+arctan(B/A))
Asinα+Bcosα=√(A²+B²)cos(α-arctan(A/B))
√3sinx+cosx
=2(√3sinx)/2+(cosx)/2)
=2(sinx·sin(π/3) +cosx·cos(π/3)
=2cos(x-π/3)=2cos(x-60°)
√3sinx+cosx
=√((√3)²;+1²;)sin(x+arctan(1/√3))
=√4;sin(x+π/3) =2sin(x+π/6) =2sin(x+30°)
3、
cos(α+β)=cosα·cosβ-sinα·sinβ
sin(α±β)=sinα·cosβ±cosα·sinβ
√2(sinx-cosx)
=√2sinx-√2cosx
=2((√2sinx)/2-(√2cosx)/2)
=2(sinx·sin(π/4)-cosx·cos(π/4) )
=-2(cosx·cos(π/4) -sinx·sin(π/4))
=-2cos(x+π/4)=-2cos(x+45°)
√2(sinx-cosx)
=2((√2sinx)/2-(√2cosx)/2)
=2(sinx·cos(π/4) -cosx·sin(π/4))
=2sin(x-π/4)=2sin(x-45°)
4
cos(α+β)=cosα·cosβ-sinα·sinβ
sin(α±β)=sinα·cosβ±cosα·sinβ
√2cosx-√6sinx
=√2(cosx-√3sinx)
=2√2((1/2)cosx-(√3/2)sinx)
=2√2(cosx·cos(π/3)-sinx·sin(π/3))
=2√2(cosx·cos(π/3)-sinx·sin(π/3))
=-2√2cos(x+π/3)=-2cos(x+60°)
√2cosx-√6sinx
=√2(cosx-√3sinx)
=2√2((1/2)cosx-(√3/2)sinx)
=2√2(cosx·sin(π/6)-sinx·cos(π/6))
=2√2sin(π/6-x)=2√2sin(30°-x)
5.已知sin(α-β)·cosα-cos(β-α)sinα=3／5,β是第三象限角,求sin（β+5π／4）的值
∵sin(α-β)·cosα-cos(β-α)sinα=3/5
sin(α-β)·cosα-cos(-(α-β))sinα=3/5
∴sin(α-β)·cosα-cos(α-β)sinα=3/5
根据sin(A-B) =sinA·cosB-cosA·sinB
sin(α-β-α)=3/5
sin(-β)=3/5
-sinβ=3/5
sinβ = -3/5
∵β是第三象限角
∴cosβ = -4/5
sin(β+5π/4)
=sin(β+π+π/4)
=－sin(β+π/4)
=－(sinβ·cosπ/4+cosβ·sinπ/4)
=－√2/2(sinβ+cosβ·)
=－√2/2(-3/5 -4/5)
=√2/2(7/5 )=7√2/10