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формулы: ![\sin \alpha +\sin \beta =2\sin \frac{ \alpha + \beta }{2} \cdot \cos \frac{ \alpha - \beta }{2} \sin \alpha +\sin \beta =2\sin \frac{ \alpha + \beta }{2} \cdot \cos \frac{ \alpha - \beta }{2}](https://tex.z-dn.net/?f=%5Csin++%5Calpha+%2B%5Csin++%5Cbeta+%3D2%5Csin++%5Cfrac%7B+%5Calpha+%2B+%5Cbeta+%7D%7B2%7D+%5Ccdot+%5Ccos++%5Cfrac%7B+%5Calpha+-+%5Cbeta+%7D%7B2%7D+)
![\sin 3x+\sin 5x+2\sin \frac{x}{2} =1 \\ (\sin 5x+\sin 3x)+2\cdot \frac{1-\cos x}{2} =1 \\ \sin 5x+\sin 3x-\cos x=0 \\ 2\sin \frac{5x+3x}{2} \cos \frac{5x-3x}{2} -\cos x=0 \\ 2\sin 4x\cos x-\cos x=0 \\ \cos x(2\sin 4x-1)=0 \\ \\ \cos x=0 \\ x_1= \frac{\pi}{2} + \pi n,n \in Z \\ \\ \sin 4x= \frac{1}{2} \\ 4x=(-1)^k\cdot \frac{\pi}{6} +\pi n,n \in Z \\ x_2=(-1)^k\cdot \frac{\pi}{24} + \frac{\pi n}{4} ,n \in Z \sin 3x+\sin 5x+2\sin \frac{x}{2} =1 \\ (\sin 5x+\sin 3x)+2\cdot \frac{1-\cos x}{2} =1 \\ \sin 5x+\sin 3x-\cos x=0 \\ 2\sin \frac{5x+3x}{2} \cos \frac{5x-3x}{2} -\cos x=0 \\ 2\sin 4x\cos x-\cos x=0 \\ \cos x(2\sin 4x-1)=0 \\ \\ \cos x=0 \\ x_1= \frac{\pi}{2} + \pi n,n \in Z \\ \\ \sin 4x= \frac{1}{2} \\ 4x=(-1)^k\cdot \frac{\pi}{6} +\pi n,n \in Z \\ x_2=(-1)^k\cdot \frac{\pi}{24} + \frac{\pi n}{4} ,n \in Z](https://tex.z-dn.net/?f=%5Csin+3x%2B%5Csin+5x%2B2%5Csin++%5Cfrac%7Bx%7D%7B2%7D+%3D1+%5C%5C+%28%5Csin+5x%2B%5Csin+3x%29%2B2%5Ccdot++%5Cfrac%7B1-%5Ccos+x%7D%7B2%7D+%3D1+%5C%5C+%5Csin+5x%2B%5Csin+3x-%5Ccos+x%3D0+%5C%5C+2%5Csin++%5Cfrac%7B5x%2B3x%7D%7B2%7D+%5Ccos++%5Cfrac%7B5x-3x%7D%7B2%7D+-%5Ccos+x%3D0+%5C%5C+2%5Csin+4x%5Ccos+x-%5Ccos+x%3D0+%5C%5C+%5Ccos+x%282%5Csin+4x-1%29%3D0+%5C%5C++%5C%5C+%5Ccos+x%3D0+%5C%5C+x_1%3D+%5Cfrac%7B%5Cpi%7D%7B2%7D+%2B+%5Cpi+n%2Cn+%5Cin+Z+%5C%5C++%5C%5C+%5Csin+4x%3D+%5Cfrac%7B1%7D%7B2%7D++%5C%5C+4x%3D%28-1%29%5Ek%5Ccdot++%5Cfrac%7B%5Cpi%7D%7B6%7D+%2B%5Cpi+n%2Cn+%5Cin+Z+%5C%5C+x_2%3D%28-1%29%5Ek%5Ccdot++%5Cfrac%7B%5Cpi%7D%7B24%7D+%2B+%5Cfrac%7B%5Cpi+n%7D%7B4%7D+%2Cn+%5Cin+Z)
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