1)3cos2x+sin^2x+5sinx cosx=0 (cos2x=cos^2x -sin^2x)
3cos^2x -3 sin^2x+sin^2x+5sinx cosx=0 (/cos^2x )
3+5tgx-2tgx^2x=0
2tgx^2x-5tgx-3=0
Пусть tgx=t
2t^2-5t-3=0
D=49
t1=3
tgx=3
x=arctg3+ πn, n∈Z
t2=1/2
tgx=1/2
x=arctg1/2+ πn, n∈Z
2)sin 7x - sin x= cos 4 x (sinα-sinβ=2sin(α-β)/2cos(α+β)/2)
2sin3xcos4x-cos4x=0
cos4x(2sin3x-1)=0
1. cos 4x=0
4x=π/2+πn, n∈Z
x=π/8+πn/4, n∈Z
2. 2sin3x-1=0
sin3x=1/2
3x=(-1)^k arsin1/2+ πk, k∈Z
x=(-1)^k π/18+ πk/3, k∈Z
3)cosx + cos 3 x=4cos2x (cosα+cosβ=2cos(α+β)/2cos(α-β)/2)
2cos2xcosx=4cos2x
cos2xcosx=2cos2x
cos2x(cosx-2)=0
1. cos2x=0
2x=π/2+πn, n∈Z
x=π/4+πn/2, n∈Z
2. cosx≠2
4)1-sin x cos x +2 cos^2x=0 (1=cos^2x +sin^2x)
cos^2x +sin^2x-sin x cos x +2 cos^2x=0 (/cos^2x )
1 + tg^2x - tgx +2 = 0
tg^2x - tgx+3=0
D=1-12≠ -11
5)arcsin 1 /√2 - 4arcsin 1=0
(arcsin1=π/2, cos(arcsin1)=cosπ/2=0).
6)arccos(-1) - arcsin (-1)=
=π-arccosx - arcsinx =
=π-(arccosx + arcsinx)= (arccosx + arcsinx=π/2)
=π- π/2= π/2
7)4arctg(-1)+3 arctg√ 3 =
=4(-π/4)+3(π/6)= (tg45=1 (45 °=π/4 ). tg(-45)=-1. arctg(-1)=-45=-π/4
=-π+π/2= -π/2 tg30=√3 arctg√3=30°=π/6)