1
a)sin765=sin(720+45)=sin45=√2/2
b)cos19π/6=cos(3π+π/6)=-cosπ/6=-√3/2
2
cosa=-√(1-sin²a)=-√(1-0,09)=-√0,91
3
1)cosacosb+sinasinb-cosacosb+sinasinb=2sinasinb
2)(-sina-cosa)/(1-2cos²a)=(sina+cosa)/cos2a=
=(sina+cosa)/(cosa-sina)(cosa+sina)=1/(cosa-sina)
4
sinxcos3x+cosxsin3x=1
sin(x+3x)=1
sin4x=1
4x=π/2+2πn,n∈z
x=π/8+πn/2,n∈z
5
(sina/cosa+cosa/sina)*2sin²2a=(sin²a+cos²a)/sinacosa*2sin²2a=
=2/sin2a*2sin²2a=4sin2a
6
a)sinx=1/√2
x=(-1)^n*π/4+πn,n∈z
b)tgx/2=√3
x/2=π/3+πn,n∈z
x=2π/3+2πn,n∈z
7
cosx/2=1/2
1)x/2=-π/3+2πn,n∈z⇒x=-2π/3+4πn
0≤-2π/3+4πn≤4π
0≤-2+12n≤12
2≤12n≤14
1/6≤n≤7/6
n=1⇒x=-2π/3+4π=10π/3
2)x/2=π/3+2πk,k∈z⇒x=2π/3+4πk,k∈z
0≤2π/3+4πk≤4π
0≤2+12k≤12
-2≤12k≤10
-1/6≤k≤5/6
k=0⇒x=2π/3
8
1)sinx(sinx-1)=0
sinx=0⇒x=πn,n∈z
sinx=1⇒x=π/2+2πk,k∈z
2)cosx=a
10a²+3a-1=0
D=9+40=49
a1=(-3-7)/20=-1/2⇒cosx=-1/2⇒x=+-2π/3+2πn,n∈z
a2=(-3+7)/20=1/5⇒cosx=1/5⇒x=+-arccos1/5+2πk,k∈z
3)10sinx/2cosx/2+cos²x/2-sin²x/2-5sin²x/2-5cos²x/2=0/cos²x/2
6tg²x/2-10tgx/2+4=0
tgx/2=a
6a²-10a+4=0
D=100-96=4
a1=(10-2)/12=2/3⇒tgx/2=2/3⇒x/2=arctg2/3+πn⇒x=2arctg2/3+πn,n∈z
a2=(10+2)/12=1⇒tgx/2=1⇒x/2=π/4+πk,k∈z⇒x=π/2+2πk,k∈z
4)(1-cos2x)²/4+(1+cos2x)²/4=sin²2x
1-2cos2x+cos²2x+1+2cos2x+cos²2x=16sin²2x
2+2cos²2x=16sin²2x
1+cos²2x-8sin²2x=0
8(1-cos4x)/2-(1+cos4x)/2=1
8-8cos4x-1-cos4x=2
9cos4x=5
cos4x=5/9
4x=+-arccos5/9+2πn,n∈z
x=+-1/4*arccos5/9+πn/2,n∈z