1.
1) 3 5/8+1 2/3=(3+1)+(5/8+2/3)=4+5*3/24+2*8/24=4+15/24+16/24=4+31/24=4 31/24=5 7/24
2) 4 4/9-2 5/6=(4-2)+(4/9-5/6)=2+4*2/18-5*3/18=2+8/18-15/18=2-7/18=1 18/18-7/18=1 11/18
3) 6 7/12+(5 3/40-4 8/15)=6 7/12+((5-4)+(3/40-8/15))=6 7/12+(1+3*3/120-8*8/120)=6 7/12+(1+9/120-64/120)=6 7/12+(1-55/120)=6 7/12+(1-11/24)=6 7/12+(24/24-11/24)=6 7/12+13/24=6+(7*2/24+13/24)=6+14/24+13/24=6+27/24=6+1 3/24=6+1 1/8=7 1/8
2.
1) 5 4/5+1 1/2=(5+1)+(4/5+1/2)=6+4*2/10+1*5/10=6+8/10+5/10=6+13/10=6+1 3/10=7 3/10 - масса другой детали
2) 5 4/5+7 3/10=(5+7)+(4/5+3/10)=12+(4*2/10+3/10)=12+(8/10+3/10)=12+11/10=12+1 1/10=13 1/10 - масса двух деталей вместе
3.
1) 5/6+2 3/4=2+(5/6+3/4)=2+(5*2/12+3*3/12)=2+(10/12+9/12)=2+19/12=2+1 7/12=3 7/12 - рассчитывал потратить садовник на работу
2) 3 7/12-1 1/4=(3-1)+(7/12-1/4)=2+(7/12-1*3/12)=2+(7/12-3/12)=2+4/12=2+1/3=2 1/3 - времени ушло у садовника на всю работу
4.
5 5/33+y=8 3/44
y=8 3/44-5 5/33
y=(8-5)+(3/44-5/33)
y=3+3*3/132-5*4/132
y=3+9/132-20/132
y=3-11/132
y=3-1/12
y=2 12/12-1/12
y=2 11/12
5.
60=1*60
60=4*15
60=20*3
60=12*5
4x=π+2πn, x=π/4+πn/2,n∈Z,
2). sin(4x-π/3)=1/2
4x-π/3=(-1)ⁿπ/6+πn, n∈Z, 4x=π/3+ (-1)ⁿπ/6+πn ,
x= 4π/3+ (-1)ⁿ 2π/3+4πn , n∈Z.
3).2sin²(x/2)=1, sin²(x/2)=1/2, sin(x/2)=1/√2, sin(x/2)=-1/√2
x/2=(-1)ⁿπ/4+πn,n∈Z, x/2=(-1)ⁿ⁺¹/π4+ πn, n∈Z
x= (-1)ⁿπ/2+2πn,n∈Z, x= (-1)ⁿπ/2+2πn,n∈Z,
4). cos4x-cos5x=0
cos4x-cos5x=-2sin(4x+5x)/2·sin(4x-5x)/2=0
-2sin(4,5x)·sin(-0,5x) =2sin 4,5x·sin0,5x=0, sin 4,5x·sin0,5x=0
sin4,5x=0, sin0,5x=0
4,5x=πn 0,5x=πn
9x/2=πn x/2=πn/ n∈Z
x=2πn/9 x=2πn, n∈Z
5.cosx- √cosx =0, √cosx(√cosx-1)=0
√cosx=0 √cosx=1
cosx=0 cosx=1
x=π/2 +2πn , n∈Z x=2πn , n∈Z
6. cos2x*cos(x+π/6)+sin2x*sin(x+π/6)=0
Воспользуемся формулой:
cosαcosβ+sinαsinβ= cos(α-β)
cos2x*cos(x+π/6)+sin2x*sin(x+π/6)=cos(2x-(x+π/6))=cos(2x-x-π/6)=0
cos(x-π/6)=0, x-π/6=π/2+2πn, x=π/6+π/2+2πn,n∈Z
x=(π+3π)/6+2πn,n∈Z, x=4π/6+ +2πn,n∈Z, x=2π/3+2πn.n∈Z