1. sin(3x)=1/2;
3x=(-1)^k*pi/6 +pik
x=(-1)^k*pi/18 +pik/3,
2. cos(x/2)=-sgrt3/2;
x/2=+-5pi/6+2pik
x=+-5pi/3+4pik/
3. ctg(x-pi/4)=sgrt3;
x-pi/4=pi/6+pik;
x=pi/6+pi/4+pik;
x=5pi/12 +pik.
4. 2cos^2x-cosx-1=0
cosx=t;
2t^2-t-1=0
t1=1: cosx=1; x=2pik;
t2=-1/2; cosx=-1/2; x=+-pi/3+2pik/
5. 3tgx-2/tgx - 1=0
3tg^2x-tgx-2=0
tgx=t
3t^2-t-2=0
t1=1; tgx=1; x=pi/4+pik
t2=-2/3; tgx=-2/3; x=-arctg(2/3)+pik/
6. 1-2sin^2(x/3) +5sin(x/3)+2=0;
2sin^2(x/3)-5sin(x/3)-3=0
sin(x/3)=t;
2t^2-5t-3=0
t1=-1; sin(x/3)=-1; x/3=-pi/2+2pik; x=-3pi/2+6pik=pi/2+6pik;
t2=3 >1 Нет решений. ОТвет:x=pi/2+6pik
По формулам приведения:
![sin( \pi +2 \alpha )=-sin2 \alpha \\ \\ tg( \frac{ \pi }{2}+ \alpha )=-ctg \alpha \\ \\ \frac{1-cos2 \alpha }{sin( \pi +2 \alpha )}\ tg( \frac{ \pi }{2}+ \alpha )= \frac{1-cos2 \alpha }{-sin2 \alpha }\cdot (-ctg \alpha )= \\ = \frac{1-cos2 \alpha }{-2sin \alpha\cdot cos \alpha }\cdot ( -\frac{cos \alpha }{sin \alpha } )= \\ = \frac{sin ^{2} \alpha +cos ^{2} \alpha -cos ^{2} \alpha +sin ^{2} \alpha }{2sin ^{2} \alpha }= \frac{2sin ^{2} \alpha }{2sin ^{2} \alpha }} =1](https://tex.z-dn.net/?f=sin%28+%5Cpi+%2B2+%5Calpha+%29%3D-sin2+%5Calpha++%5C%5C++%5C%5C+tg%28+%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2B+%5Calpha+%29%3D-ctg+%5Calpha++%5C%5C++%5C%5C++%5Cfrac%7B1-cos2+%5Calpha+%7D%7Bsin%28+%5Cpi+%2B2+%5Calpha+%29%7D%5C+tg%28+%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2B+%5Calpha+%29%3D++++%5Cfrac%7B1-cos2+%5Calpha+%7D%7B-sin2+%5Calpha+%7D%5Ccdot+%28-ctg+%5Calpha+%29%3D+%5C%5C+%3D++++%5Cfrac%7B1-cos2+%5Calpha+%7D%7B-2sin+%5Calpha%5Ccdot+cos+%5Calpha++%7D%5Ccdot+%28+-%5Cfrac%7Bcos+%5Calpha+%7D%7Bsin+%5Calpha+%7D++%29%3D+%5C%5C+%3D++++%5Cfrac%7Bsin+%5E%7B2%7D++%5Calpha+%2Bcos+%5E%7B2%7D+%5Calpha++-cos+%5E%7B2%7D++%5Calpha+%2Bsin+%5E%7B2%7D+%5Calpha++%7D%7B2sin+%5E%7B2%7D++%5Calpha+%7D%3D+%5Cfrac%7B2sin+%5E%7B2%7D++%5Calpha+%7D%7B2sin+%5E%7B2%7D++%5Calpha+%7D%7D+%3D1++)
![cos( \frac{3 \pi }{2}-2 \alpha)=-sin2 \alpha \\ \\ ctg( \pi + \alpha )=ctg \alpha](https://tex.z-dn.net/?f=cos%28+%5Cfrac%7B3+%5Cpi+%7D%7B2%7D-2+%5Calpha%29%3D-sin2+%5Calpha++%5C%5C++%5C%5C+ctg%28+%5Cpi+%2B+%5Calpha+%29%3Dctg+%5Calpha++)
![\frac{cos( \frac{3 \pi }{2}+ \alpha ) }{1+cos2 \alpha } \cdot ctg( \pi + \alpha )=\frac{-sin \alpha }{cos ^{2} \alpha +sin ^{2} \alpha +cos ^{2} \alpha-sin ^{2} \alpha } \cdot \frac{cos \alpha }{sin \alpha } = -\frac{1}{2cos \alpha }](https://tex.z-dn.net/?f=+%5Cfrac%7Bcos%28+%5Cfrac%7B3+%5Cpi+%7D%7B2%7D%2B+%5Calpha+%29+%7D%7B1%2Bcos2+%5Calpha+%7D+%5Ccdot+ctg%28+%5Cpi+%2B+%5Calpha+%29%3D%5Cfrac%7B-sin+%5Calpha+%7D%7Bcos+%5E%7B2%7D+%5Calpha+%2Bsin+%5E%7B2%7D+%5Calpha+++%2Bcos+%5E%7B2%7D++%5Calpha-sin+%5E%7B2%7D+%5Calpha+++%7D+%5Ccdot++%5Cfrac%7Bcos+%5Calpha+%7D%7Bsin+%5Calpha+%7D+%3D+-%5Cfrac%7B1%7D%7B2cos+%5Calpha+%7D+)
у=х²-2х-3
у=0- уравнение оси Ох
х²-2х-3=0
D=4+12=16
x=(2-4)/2=-1 или х=(2+4)/2=3
(-1;0) (3;0)- точки пересечения с осью Ох
х=0- уравнение оси Оу
у=0²-2·0-3=-3
(0;-3) - точка пересечения с осью Оу
у=х²+х-2
у=0- уравнение оси Ох
х²+х-2=0
D=1+8=9
x=(-1-3)/2=-2 или х=(-1+3)/2=1
(-2;0) (1;0)- точки пересечения с осью Ох
х=0- уравнение оси Оу
у=0²+0-2=-2
(0;-2) - точка пересечения с осью Оу