1
1)cosx=1/√2
x=+-π/4+2πk,k∈z
2)tg2x=-√3/3
2x=-π/6+2πk
x=-π/12+πk/2,k∈z
2
x/3=7π/6+2πk,k∈z U x=11π/6+2πn,n∈z
0≤7π/6+2πk≤3π
0≤7+12k≤18
-7≤12k≤11
-7/12≤k≤11/12
k=0⇒x=7π/6
0≤11π/6+2πn≤3π
0≤11+12n≤18
-11≤12n≤7
-11/12≤n≤7/12
n=0⇒x=11π/6
3
1)cosx(3-cosx)=0
cosx=0⇒x=π/2+πk,k∈z
cosx=3>1 нет решения
2)sinx=a
6a²-a-1=0
D=1+24=25
a1=(1-5)/12=-1/3⇒sinx=-1/3⇒x=(-1)^(n+1)*arcsin1/3+πn,n∈z
a2=(1+5)/12=1/2⇒sinx=1/2⇒x=(-1)^k*π/6+πk,k∈z
3)8sin(x/2)cos(x/2)+5cos²(x/2)-5sin²(x/2)-4sin²(x/2)-4cos²(x/2)=0/cos²(x/2)
9tg²(x/2)-8tg(x/2)-1=0
tg(x/2)=a
9a²-8a-1=0
D=64+36=100
a1=(8-10)/18=-1/9⇒tg(x/2)=-1/9⇒x/2=-arctg1/9+πn⇒
x=-2arctg1/9+2πn,n∈z
a2=(8+10)/18=1⇒tg(x/2)=1⇒x/2=π/4+πk⇒x=π/2+2πk,k∈z
4)(1-cos2x)²/4+(1+cos2x)²/4=cos²2x+1/4
1-2cos2x+cos²2x+1+2cos2x+cos²2x=4cos²2x+1
2+2cos²2x-4cos²2x-1=0
2cos²2x-1=0
cos4x=0
4x=π/2+πk,k∈z
x=π/8+πk/4,k∈z