作业帮如何直接得出cos36-cos72=1/2 cos36*cos72=1/4?
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/11 05:17:06
如何直接得出cos36-cos72=1/2 cos36*cos72=1/4?
如何直接得出cos36-cos72=1/2 cos36*cos72=1/4?
如何直接得出cos36-cos72=1/2 cos36*cos72=1/4?
cos36-cos72
解1.Cos36-cos72
=(sin36cos36-sin36cos72)/sin36
=-(sin108-sin36-sin72)/2sin36
=-(sin72-sin36-sin72)/2sin36
=sin36/2sin36
=1/2
解2.cos36-cos72
=cos(54-18)-cos(54+18)
=cos54cos18+sin54sin18-[cos54cos18-sin54sin18]
=2sin54sin18
=2cos36sin18
=2cos36sin18cos18/cos18
=cos36sin36/cos18
=sin72/(2cos18)
=sin72/(2sin72)=1/2
解3.做单位向量OA=(cos0,sin0),OB=(cos72,sin72),OC=(cos144,sin144)
OD=(cos216,sin216),OE=(cos288,sin288),
OA,OB,OC,OD,OE是均匀分布的5个单位向量
所以OA+OB+OC+OD+OE=(0,0)
OA+OB+OC+OD+OE
=(cos0+cos72+cos144+cos216+cos288,sin0+sin72+sin144+sin216+sin288)
所以cos0+cos72+cos144+cos216+cos288=0
即,1+cos72-cos36-cos36+cos72=0
2(cos36-cos72)=1
cos36-cos72=1/2
解4.1）先解出sin18,设sin18=x.
因为cos36=sin54,注意36=18*2,54=18*3,由2 倍角和3倍角公式,得
1-2x^2=3x-4x^3,即4x^3-2x^2-3x+1=（x-1)(4x^2+2x-1)=0,
由于sin18≠1,解4x^2+2x-1=0得x=（√5-1）/4=sin18
2)x=cos36-cos72=1-2(sin18)^2-sin18=0.5
解5.x=cos36-cos72
x*2sin36=2sin36cos36-2sin36cos72
=sin72-(sin108-sin36)
=sin36
故x=0.5
解6.cos36-cos72
=2sin54*sin18
=2cos36*cos72
=4sin36*cos36*cos72/(2sin36)
=2sin72*cos72/(2sin36)
=sin144/(2sin36)
=1/2
Cos36*Cos72
=Cos36*Cos72*Sin72/Sin72
=Cos36*Sin144/4(Sin36*Cos36)
=Sin144/4Sin36
=1/4