Question

Решить тригонометрические уравнения 1) cos^2(x)+sinx-1=0 2) sin5x*cosx+sinx*cos5x=1 3) 3sin-9cosx=0 4) sin2x+sinx=0 5) 3sin^2(7/2x)-cos^2(7/2)x=-1/2

Answer 1

2) формула sin(a+b)
sin(5x+x)=1
sin6x=1
6x=pi/2+2pik
x=pi/12+pik/3
4)2sinxcosx+sinx=0
sinx(2cosx+1)=0
sinx=0 x=pik
cosx=-1/2 x=pi-pi/3=2pi/3

Answer 2

1) cos^2(x)+sinx-1=0
1-sin²x+sinx-1=0
sinx-sin²x=0
sinx(1-sinx)=0
sinx=0⇒x=πn,n∈z
sinx=1⇒x=π/2+2πk,k∈z
2) sin5x*cosx+sinx*cos5x=1
sin(5x+x)=1
sin6x=1⇒6x=π/2+2πn⇒x=π/12+πn/3,n∈z
3) 3sin-9cosx=0/cosx
3tgx-9=0
3tgx=9
tgx=3
x=arctg3+πn,n∈z
4) sin2x+sinx=0
2sinxcosx+sinx=0
sinx(2cosx+1)=0
sinx=0⇒x=πn,n∈z
cosx=-1/2⇒x=+-2π/3+2πk,k∈z
 5) 3sin^2(7/2x)-cos^2(7/2)x=-1/2
3/2*(1-cos7x)-1/2*(1+cos7x)=-1/2
3-3cos7x-1-cos7x=-1
4cos7x=3
cos7x=3/4
7x=+-arccos3/4+2πn
x=+-1/7*arccos0,75+2πn/7,n∈z