Ответы
Ответ дал:
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№ 1.
![frac{4ctgx}{1+ctg^{2}x}+sin^{2}2x+1=0 frac{4ctgx}{1+ctg^{2}x}+sin^{2}2x+1=0](https://tex.z-dn.net/?f=+frac%7B4ctgx%7D%7B1%2Bctg%5E%7B2%7Dx%7D%2Bsin%5E%7B2%7D2x%2B1%3D0)
![frac{4cosx}{sinx(1+ frac{cos^{2}x}{sin^{2}x})}+4sin^{2}x*cos^{2}x+1=0 frac{4cosx}{sinx(1+ frac{cos^{2}x}{sin^{2}x})}+4sin^{2}x*cos^{2}x+1=0](https://tex.z-dn.net/?f=+frac%7B4cosx%7D%7Bsinx%281%2B+frac%7Bcos%5E%7B2%7Dx%7D%7Bsin%5E%7B2%7Dx%7D%29%7D%2B4sin%5E%7B2%7Dx%2Acos%5E%7B2%7Dx%2B1%3D0)
![frac{4cosx}{sinx(frac{sin^{2}x+cos^{2}x}{sin^{2}x})}+4sin^{2}x*cos^{2}x+1=0 frac{4cosx}{sinx(frac{sin^{2}x+cos^{2}x}{sin^{2}x})}+4sin^{2}x*cos^{2}x+1=0](https://tex.z-dn.net/?f=+frac%7B4cosx%7D%7Bsinx%28frac%7Bsin%5E%7B2%7Dx%2Bcos%5E%7B2%7Dx%7D%7Bsin%5E%7B2%7Dx%7D%29%7D%2B4sin%5E%7B2%7Dx%2Acos%5E%7B2%7Dx%2B1%3D0)
![4cosx*sinx+4sin^{2}x*cos^{2}x+1=0 4cosx*sinx+4sin^{2}x*cos^{2}x+1=0](https://tex.z-dn.net/?f=4cosx%2Asinx%2B4sin%5E%7B2%7Dx%2Acos%5E%7B2%7Dx%2B1%3D0)
Замена: sinx*cosx=t
![4t^{2}+4t+1=(2t+1)^{2}=0 4t^{2}+4t+1=(2t+1)^{2}=0](https://tex.z-dn.net/?f=4t%5E%7B2%7D%2B4t%2B1%3D%282t%2B1%29%5E%7B2%7D%3D0)
![t=-0.5 t=-0.5](https://tex.z-dn.net/?f=t%3D-0.5)
Вернемся к замене:
![sinx*cosx=-0.5 sinx*cosx=-0.5](https://tex.z-dn.net/?f=sinx%2Acosx%3D-0.5)
![2*0.5*sinx*cosx=-0.5 2*0.5*sinx*cosx=-0.5](https://tex.z-dn.net/?f=2%2A0.5%2Asinx%2Acosx%3D-0.5)
![2sinx*cosx=-1 2sinx*cosx=-1](https://tex.z-dn.net/?f=2sinx%2Acosx%3D-1)
![sin2x=-1 sin2x=-1](https://tex.z-dn.net/?f=sin2x%3D-1)
, k∈Z
, k∈Z
№ 2.
![cos4x+2cos^{2}x-1=0 cos4x+2cos^{2}x-1=0](https://tex.z-dn.net/?f=cos4x%2B2cos%5E%7B2%7Dx-1%3D0)
![2cos^{2}x-1+2cos^{2}x-1=0 2cos^{2}x-1+2cos^{2}x-1=0](https://tex.z-dn.net/?f=2cos%5E%7B2%7Dx-1%2B2cos%5E%7B2%7Dx-1%3D0)
![4cos^{2}x-2=0 4cos^{2}x-2=0](https://tex.z-dn.net/?f=4cos%5E%7B2%7Dx-2%3D0)
![2cos^{2}x=1 2cos^{2}x=1](https://tex.z-dn.net/?f=2cos%5E%7B2%7Dx%3D1)
![cos^{2}x=0.5 cos^{2}x=0.5](https://tex.z-dn.net/?f=cos%5E%7B2%7Dx%3D0.5)
a)![cosx= frac{ sqrt{2}}{2} cosx= frac{ sqrt{2}}{2}](https://tex.z-dn.net/?f=cosx%3D+frac%7B+sqrt%7B2%7D%7D%7B2%7D+)
, k∈Z
b)![cosx=- frac{ sqrt{2}}{2} cosx=- frac{ sqrt{2}}{2}](https://tex.z-dn.net/?f=cosx%3D-+frac%7B+sqrt%7B2%7D%7D%7B2%7D+)
, k∈Z
Объединим решения в одно:
, k∈Z
№ 3.
![4sinx+cosx=4 4sinx+cosx=4](https://tex.z-dn.net/?f=4sinx%2Bcosx%3D4)
![4*2sin frac{x}{2}*cosxfrac{x}{2}+cos^{2}frac{x}{2}-sin^{2}frac{x}{2}=4sin^{2}frac{x}{2}+4cos^{2}frac{x}{2} 4*2sin frac{x}{2}*cosxfrac{x}{2}+cos^{2}frac{x}{2}-sin^{2}frac{x}{2}=4sin^{2}frac{x}{2}+4cos^{2}frac{x}{2}](https://tex.z-dn.net/?f=4%2A2sin+frac%7Bx%7D%7B2%7D%2Acosxfrac%7Bx%7D%7B2%7D%2Bcos%5E%7B2%7Dfrac%7Bx%7D%7B2%7D-sin%5E%7B2%7Dfrac%7Bx%7D%7B2%7D%3D4sin%5E%7B2%7Dfrac%7Bx%7D%7B2%7D%2B4cos%5E%7B2%7Dfrac%7Bx%7D%7B2%7D)
![8sin frac{x}{2}*cosxfrac{x}{2}-3cos^{2}frac{x}{2}-5sin^{2}frac{x}{2}=0 8sin frac{x}{2}*cosxfrac{x}{2}-3cos^{2}frac{x}{2}-5sin^{2}frac{x}{2}=0](https://tex.z-dn.net/?f=8sin+frac%7Bx%7D%7B2%7D%2Acosxfrac%7Bx%7D%7B2%7D-3cos%5E%7B2%7Dfrac%7Bx%7D%7B2%7D-5sin%5E%7B2%7Dfrac%7Bx%7D%7B2%7D%3D0)
![8tgfrac{x}{2}-3-5tg^{2}frac{x}{2}=0 8tgfrac{x}{2}-3-5tg^{2}frac{x}{2}=0](https://tex.z-dn.net/?f=8tgfrac%7Bx%7D%7B2%7D-3-5tg%5E%7B2%7Dfrac%7Bx%7D%7B2%7D%3D0)
![5tg^{2}frac{x}{2}-8tgfrac{x}{2}+3=0 5tg^{2}frac{x}{2}-8tgfrac{x}{2}+3=0](https://tex.z-dn.net/?f=5tg%5E%7B2%7Dfrac%7Bx%7D%7B2%7D-8tgfrac%7Bx%7D%7B2%7D%2B3%3D0)
Замена: tg(x/2)=t
![5t^{2}-8t+3=0, D=64-4*3*5=4 5t^{2}-8t+3=0, D=64-4*3*5=4](https://tex.z-dn.net/?f=5t%5E%7B2%7D-8t%2B3%3D0%2C+D%3D64-4%2A3%2A5%3D4)
![t_{1}= frac{8-2}{10}=frac{3}{5}=0.6 t_{1}= frac{8-2}{10}=frac{3}{5}=0.6](https://tex.z-dn.net/?f=t_%7B1%7D%3D+frac%7B8-2%7D%7B10%7D%3Dfrac%7B3%7D%7B5%7D%3D0.6)
![t_{2}= frac{8+2}{10}=1 t_{2}= frac{8+2}{10}=1](https://tex.z-dn.net/?f=t_%7B2%7D%3D+frac%7B8%2B2%7D%7B10%7D%3D1)
Вернемся к замене:
a)![tg frac{x}{2}=0.6 tg frac{x}{2}=0.6](https://tex.z-dn.net/?f=tg+frac%7Bx%7D%7B2%7D%3D0.6)
, k∈Z
, k∈Z
b)![tgfrac{x}{2}=1 tgfrac{x}{2}=1](https://tex.z-dn.net/?f=tgfrac%7Bx%7D%7B2%7D%3D1)
, k∈Z
, k∈Z
Замена: sinx*cosx=t
Вернемся к замене:
№ 2.
a)
b)
Объединим решения в одно:
№ 3.
Замена: tg(x/2)=t
Вернемся к замене:
a)
b)
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