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1)
![sin(x- frac{ pi }{3})*cos(x- frac{ pi }{6} )=1 sin(x- frac{ pi }{3})*cos(x- frac{ pi }{6} )=1](https://tex.z-dn.net/?f=sin%28x-+frac%7B+pi+%7D%7B3%7D%29%2Acos%28x-+frac%7B+pi+%7D%7B6%7D++%29%3D1)
![frac{1}{2} [sin(x- frac{ pi }{3}+x- frac{ pi }{6} )+sin(x- frac{ pi }{3} -x+frac{ pi }{6} )]=1 frac{1}{2} [sin(x- frac{ pi }{3}+x- frac{ pi }{6} )+sin(x- frac{ pi }{3} -x+frac{ pi }{6} )]=1](https://tex.z-dn.net/?f=+frac%7B1%7D%7B2%7D+%5Bsin%28x-+frac%7B+pi+%7D%7B3%7D%2Bx-+frac%7B+pi+%7D%7B6%7D+%29%2Bsin%28x-+frac%7B+pi+%7D%7B3%7D+-x%2Bfrac%7B+pi+%7D%7B6%7D++%29%5D%3D1)
![frac{1}{2} [sin(2x- frac{ pi }{2})+sin(- frac{ pi }{6} )]=1 frac{1}{2} [sin(2x- frac{ pi }{2})+sin(- frac{ pi }{6} )]=1](https://tex.z-dn.net/?f=+frac%7B1%7D%7B2%7D+%5Bsin%282x-+frac%7B+pi+%7D%7B2%7D%29%2Bsin%28-+frac%7B+pi+%7D%7B6%7D++%29%5D%3D1)
![-sin( frac{ pi }{2}-2x)-sin frac{ pi }{6}=2 -sin( frac{ pi }{2}-2x)-sin frac{ pi }{6}=2](https://tex.z-dn.net/?f=-sin%28+frac%7B+pi+%7D%7B2%7D-2x%29-sin+frac%7B+pi+%7D%7B6%7D%3D2)
![sin( frac{ pi }{2}-2x)+sin frac{ pi }{6}=-2 sin( frac{ pi }{2}-2x)+sin frac{ pi }{6}=-2](https://tex.z-dn.net/?f=sin%28+frac%7B+pi+%7D%7B2%7D-2x%29%2Bsin+frac%7B+pi+%7D%7B6%7D%3D-2)
![cos2x+ frac{1}{2} =-2 cos2x+ frac{1}{2} =-2](https://tex.z-dn.net/?f=cos2x%2B+frac%7B1%7D%7B2%7D+%3D-2)
![cos2x =-2.5 cos2x =-2.5](https://tex.z-dn.net/?f=cos2x+%3D-2.5)
так как![|cosx| leq 1 |cosx| leq 1](https://tex.z-dn.net/?f=%7Ccosx%7C+leq+1)
Ответ: корней нет
2)
![sin frac{x}{2}*sin frac{3x}{2} = frac{1}{2} sin frac{x}{2}*sin frac{3x}{2} = frac{1}{2}](https://tex.z-dn.net/?f=sin+frac%7Bx%7D%7B2%7D%2Asin+frac%7B3x%7D%7B2%7D+%3D+frac%7B1%7D%7B2%7D++)
![frac{1}{2} [cos( frac{x}{2} - frac{3x}{2})-cos( frac{x}{2}+ frac{3x}{2} )]= frac{1}{2} frac{1}{2} [cos( frac{x}{2} - frac{3x}{2})-cos( frac{x}{2}+ frac{3x}{2} )]= frac{1}{2}](https://tex.z-dn.net/?f=+frac%7B1%7D%7B2%7D+%5Bcos%28+frac%7Bx%7D%7B2%7D+-+frac%7B3x%7D%7B2%7D%29-cos%28+frac%7Bx%7D%7B2%7D%2B+frac%7B3x%7D%7B2%7D+%29%5D%3D+frac%7B1%7D%7B2%7D+++)
![frac{1}{2} [cos(-x)-cos2x ]= frac{1}{2} frac{1}{2} [cos(-x)-cos2x ]= frac{1}{2}](https://tex.z-dn.net/?f=+frac%7B1%7D%7B2%7D+%5Bcos%28-x%29-cos2x+%5D%3D+frac%7B1%7D%7B2%7D+++)
![cosx-cos2x =1 cosx-cos2x =1](https://tex.z-dn.net/?f=+cosx-cos2x+%3D1)
![cosx =1+cos2x cosx =1+cos2x](https://tex.z-dn.net/?f=+cosx+%3D1%2Bcos2x)
![cosx =2cos^2x cosx =2cos^2x](https://tex.z-dn.net/?f=+cosx+%3D2cos%5E2x)
![2cos^2x-cosx=0 2cos^2x-cosx=0](https://tex.z-dn.net/?f=2cos%5E2x-cosx%3D0)
![cosx(2cosx-1)=0 cosx(2cosx-1)=0](https://tex.z-dn.net/?f=cosx%282cosx-1%29%3D0)
или ![cosx= frac{1}{2} cosx= frac{1}{2}](https://tex.z-dn.net/?f=cosx%3D+frac%7B1%7D%7B2%7D+)
∈
или
∈ ![Z Z](https://tex.z-dn.net/?f=Z)
3)
![2sin ( frac{ pi }{4} +x) * sin ( frac{ pi }{4} -x) + sin^2x=0 2sin ( frac{ pi }{4} +x) * sin ( frac{ pi }{4} -x) + sin^2x=0](https://tex.z-dn.net/?f=2sin+%28+frac%7B+pi+%7D%7B4%7D+%2Bx%29+%2A+sin+%28+frac%7B+pi+%7D%7B4%7D+-x%29+%2B+sin%5E2x%3D0)
![2* frac{1}{2} [cos ( frac{ pi }{4} +x- frac{ pi }{4}+x )- cos ( frac{ pi }{4}+x+ frac{ pi }{4} -x)] + sin^2x=0 2* frac{1}{2} [cos ( frac{ pi }{4} +x- frac{ pi }{4}+x )- cos ( frac{ pi }{4}+x+ frac{ pi }{4} -x)] + sin^2x=0](https://tex.z-dn.net/?f=2%2A+frac%7B1%7D%7B2%7D+%5Bcos+%28+frac%7B+pi+%7D%7B4%7D+%2Bx-+frac%7B+pi+%7D%7B4%7D%2Bx+%29-+cos+%28+frac%7B+pi+%7D%7B4%7D%2Bx%2B+frac%7B+pi+%7D%7B4%7D++-x%29%5D+%2B+sin%5E2x%3D0)
![2* frac{1}{2} (cos 2x - cos frac{ pi }{2}) + sin^2x=0 2* frac{1}{2} (cos 2x - cos frac{ pi }{2}) + sin^2x=0](https://tex.z-dn.net/?f=2%2A+frac%7B1%7D%7B2%7D+%28cos+2x+-+cos+frac%7B+pi+%7D%7B2%7D%29+%2B+sin%5E2x%3D0)
![cos 2x - cos frac{ pi }{2} + sin^2x=0 cos 2x - cos frac{ pi }{2} + sin^2x=0](https://tex.z-dn.net/?f=cos+2x+-+cos+frac%7B+pi+%7D%7B2%7D+%2B+sin%5E2x%3D0)
![cos 2x+ sin^2x=0 cos 2x+ sin^2x=0](https://tex.z-dn.net/?f=cos+2x%2B+sin%5E2x%3D0)
![cos ^2x-sin^2x+ sin^2x=0 cos ^2x-sin^2x+ sin^2x=0](https://tex.z-dn.net/?f=cos+%5E2x-sin%5E2x%2B+sin%5E2x%3D0)
![cos ^2x=0 cos ^2x=0](https://tex.z-dn.net/?f=cos+%5E2x%3D0)
![cosx=0 cosx=0](https://tex.z-dn.net/?f=cosx%3D0)
∈ ![Z Z](https://tex.z-dn.net/?f=Z)
так как
Ответ: корней нет
2)
3)
Ответ дал:
0
Ответ ответ ответ ответ ответ
Приложения:
![](https://st.uroker.com/files/213/213825ca0b31627de0502dd3391edfad.jpg)
![](https://st.uroker.com/files/096/096004b3c6fe1df4ccf6f627e1338864.jpg)
![](https://st.uroker.com/files/9df/9df3111224281a217a82e00c9e8a76c8.jpg)
![](https://st.uroker.com/files/9a5/9a58d0c4a791bcdd11cbb59034146ad8.jpg)
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