1.1)sin2x=√3/2
a)2x=π/3+2kπ,x=π/6+kπ,k∈Z
b)2x=2π/3+2kπ, x=π/3+kπ,k∈Z
1.2)cos(x-60°)=-1/2
a)x-60°=120°+2k.180°,x=120°+60°+k.360°=180°+k.360°,k∈Z
b)x-60°=210°+2k.180°, x=210°+60°+k.360°=270°+k.360°,k∈Z
1.3)tg(x+45°)=√3
x+45°=60°+k.180°, x=60°-45°+k.180°=15°+k.180°,k∈Z
2.1)4sin²x+4sinx-3=0
sinx=z, 4z²+4z-3=0, D=16+48=64,√D=√64=8
z1=(-4+8)/2=4/8=1/2,
z2=(-4-8)/8=-12/8=-1,5 ne dact rešenie
sinx=1/2
a)x=π/6+2kπ,k∈Z
b)x=5π/6+2kπ,k∈Z
2.2)sin²x-3cosx-3=0
(1-cos²x)-3cosx-3=0
cos²x+3cosx+2=0
cosx=z, z²+3z+2=0,D=9-8=1,√D=√1=1
z1=(-3+1)/2=-2/2=-1
z2=(-3-1)/2=-4/2=-2 ne dact rešenie
cosx=-1, x=π+2kπ=(2k+1)π, k∈Z
3.1)sinx=√3cosx, sinx/cosx=√3, tgx=√3, x≠π/2+kπ
x=π/3+kπ,k∈Z
3.2)cos²x-2sinxcosx-3cos²x=0
-2sinxcosx-2cos²x=0
-2cosx(sinx+cosx)=0
a)cosx=0, x=π/2+kπ,k∈Z
b)sinx+cosx=0
x=3π/4+2kπ,k∈Z
x=7π/4+2kπ,k∈Z
4.1)sin2x-sin6x=0
2cos4xsin(-2x)=0
-2cos4xsin2x=0
cos4xsin2x=0
a)cos4x=0, 4x=π/2+kπ, x=π/8+kπ/4, k∈Z
b)sin2x=0,2x=kπ,x=kπ/2,k∈Z
4.2)cos3x+cos7x=0
2cos5xcos(-2x)=0
cos5xcos2x=0
a)cos5x=0, 5x=π/2+kπ, x=π/10+kπ/5
b)cos2x=0, 2x=π/2+kπ,x=π/4+kπ/2
4.3)2sinxcosx-sinx=0
sinx(2cosx-1)=0
a)sinx=0, x=kπ,k∈Z
b)2cosx-1=0,2cosx=1, cosx=1/2
x=π/3+2kπ
x=5π/3+2kπ, k∈Z