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6Cos²x - 7Cosx - 5 = 0
Сделаем замену Cosx = m , - 1 ≤ Cosx ≤ 1
6m² - 7m - 5 = 0
D = (- 7)² - 4 * 6 * (- 5) = 49 + 120 = 169 = 13²
![m_{1} = frac{7-13}{12} =- frac{1}{2} m_{1} = frac{7-13}{12} =- frac{1}{2}](https://tex.z-dn.net/?f=+m_%7B1%7D+%3D+frac%7B7-13%7D%7B12%7D+%3D-+frac%7B1%7D%7B2%7D+)
![m_{2} = frac{7+13}{12}= frac{5}{3}=1 frac{2}{3} textgreater 1
m_{2} = frac{7+13}{12}= frac{5}{3}=1 frac{2}{3} textgreater 1](https://tex.z-dn.net/?f=+m_%7B2%7D+%3D+frac%7B7%2B13%7D%7B12%7D%3D+frac%7B5%7D%7B3%7D%3D1+frac%7B2%7D%7B3%7D++textgreater++1+%0A+)
m₂ - не подходит
![Cosx=- frac{1}{2}\\ x_{1}= frac{2 pi }{3} +2 pi n \\ x_{2}=- frac{2 pi }{3}+2 pi n\\\- pi leq frac{2 pi }{3}+2 pi n leq 2 pi \\-1 leq frac{2}{3} +2n leq 2\\-1 frac{2}{3} leq 2n leq 1 frac{1}{3} \\- frac{5}{6} leq n leq frac{2}{3} \\n = 0 , x = frac{2 pi }{3}+2 pi *0= frac{2 pi }{3} \\\- pi leq - frac{2 pi }{3}+2 pi n leq 2 pi \\-1 leq - frac{2}{3}+2n leq 2\\- frac{1}{3} leq 2n leq 2 frac{2}{3} Cosx=- frac{1}{2}\\ x_{1}= frac{2 pi }{3} +2 pi n \\ x_{2}=- frac{2 pi }{3}+2 pi n\\\- pi leq frac{2 pi }{3}+2 pi n leq 2 pi \\-1 leq frac{2}{3} +2n leq 2\\-1 frac{2}{3} leq 2n leq 1 frac{1}{3} \\- frac{5}{6} leq n leq frac{2}{3} \\n = 0 , x = frac{2 pi }{3}+2 pi *0= frac{2 pi }{3} \\\- pi leq - frac{2 pi }{3}+2 pi n leq 2 pi \\-1 leq - frac{2}{3}+2n leq 2\\- frac{1}{3} leq 2n leq 2 frac{2}{3}](https://tex.z-dn.net/?f=Cosx%3D-+frac%7B1%7D%7B2%7D%5C%5C+x_%7B1%7D%3D+frac%7B2+pi+%7D%7B3%7D+%2B2+pi+n+%5C%5C+x_%7B2%7D%3D-+frac%7B2+pi+%7D%7B3%7D%2B2+pi+n%5C%5C%5C-+pi++leq++frac%7B2+pi+%7D%7B3%7D%2B2+pi+n+leq+2+pi+%5C%5C-1++leq++frac%7B2%7D%7B3%7D+%2B2n+leq+2%5C%5C-1+frac%7B2%7D%7B3%7D++leq+2n+leq+1+frac%7B1%7D%7B3%7D+%5C%5C-+frac%7B5%7D%7B6%7D++leq+n+leq++frac%7B2%7D%7B3%7D+%5C%5Cn+%3D+0+%2C+x+%3D++frac%7B2+pi+%7D%7B3%7D%2B2+pi+%2A0%3D+frac%7B2+pi+%7D%7B3%7D+%5C%5C%5C-+pi++leq+-+frac%7B2+pi+%7D%7B3%7D%2B2+pi+n+leq+2+pi+%5C%5C-1+leq+-+frac%7B2%7D%7B3%7D%2B2n+leq+2%5C%5C-+frac%7B1%7D%7B3%7D+leq+2n+leq+2+frac%7B2%7D%7B3%7D+++++++++)
![- frac{1}{6} leq n leq 1 frac{1}{3}\\n=0,x=- frac{2 pi }{3} +2 pi *0=- frac{2 pi }{3} \\n=1,x= - frac{2 pi }{3} +2 pi *1= frac{4 pi }{3} - frac{1}{6} leq n leq 1 frac{1}{3}\\n=0,x=- frac{2 pi }{3} +2 pi *0=- frac{2 pi }{3} \\n=1,x= - frac{2 pi }{3} +2 pi *1= frac{4 pi }{3}](https://tex.z-dn.net/?f=-+frac%7B1%7D%7B6%7D+leq+n+leq+1+frac%7B1%7D%7B3%7D%5C%5Cn%3D0%2Cx%3D-+frac%7B2+pi+%7D%7B3%7D+%2B2+pi+%2A0%3D-+frac%7B2+pi+%7D%7B3%7D+%5C%5Cn%3D1%2Cx%3D+-++frac%7B2+pi+%7D%7B3%7D+%2B2+pi+%2A1%3D+frac%7B4+pi+%7D%7B3%7D+)
Ответ:
Сделаем замену Cosx = m , - 1 ≤ Cosx ≤ 1
6m² - 7m - 5 = 0
D = (- 7)² - 4 * 6 * (- 5) = 49 + 120 = 169 = 13²
m₂ - не подходит
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