14y + 2y (6-y) = 16y,6y
7y + y * (6-y) = 4,8y^2
7y + 6y - y^2 = 24/5 y^2
13y-y^2 = 24/5 y^2
65y - 5y^2 = 24y^2
65y - 5y^2 - 24y^2 = 0
65y - 29y^2 = 0
y*(65-29y) = 0
y=0
65-29y=0
y=0
y= 65/29
y1=0
y2=65/29
6Sin²x - 7Sinx - 5 = 0
Sinx = m , - 1 ≤ m ≤ 1
6m² - 7m - 5 = 0
D = (- 7)² - 4 * 6 * (- 5) = 49 + 120 = 169 = 13²
![m_{1} =\frac{7-13}{12}=-\frac{6}{12}=-\frac{1}{2} \\\\m_{2}=\frac{7+13}{12}=\frac{20}{12}=1\frac{2}{3}](https://tex.z-dn.net/?f=m_%7B1%7D+%3D%5Cfrac%7B7-13%7D%7B12%7D%3D-%5Cfrac%7B6%7D%7B12%7D%3D-%5Cfrac%7B1%7D%7B2%7D+%5C%5C%5C%5Cm_%7B2%7D%3D%5Cfrac%7B7%2B13%7D%7B12%7D%3D%5Cfrac%7B20%7D%7B12%7D%3D1%5Cfrac%7B2%7D%7B3%7D)
Корень m₂ - не подходит так как больше единицы.
![Sinx=-\frac{1}{2}\\\\x=(-1)^{n}arcSin(-\frac{1}{2})+\pi n,n\in Z\\\\x=(-1)^{n+1}arcSin\frac{1}{2}+\pi n,n\in Z\\\\x=(-1)^{n+1}\frac{\pi }{6}+\pi n,n\in Z](https://tex.z-dn.net/?f=Sinx%3D-%5Cfrac%7B1%7D%7B2%7D%5C%5C%5C%5Cx%3D%28-1%29%5E%7Bn%7DarcSin%28-%5Cfrac%7B1%7D%7B2%7D%29%2B%5Cpi+n%2Cn%5Cin+Z%5C%5C%5C%5Cx%3D%28-1%29%5E%7Bn%2B1%7DarcSin%5Cfrac%7B1%7D%7B2%7D%2B%5Cpi+n%2Cn%5Cin+Z%5C%5C%5C%5Cx%3D%28-1%29%5E%7Bn%2B1%7D%5Cfrac%7B%5Cpi+%7D%7B6%7D%2B%5Cpi+n%2Cn%5Cin+Z)
![3\sin x+2\cos x=3](https://tex.z-dn.net/?f=3%5Csin+x%2B2%5Ccos+x%3D3)
Поделим все части на корень из суммы квадратов коэффициентов перед тригонометрическими функциями.
![\sqrt{3^2+2^2}=\sqrt{9+4}=\sqrt{13}](https://tex.z-dn.net/?f=%5Csqrt%7B3%5E2%2B2%5E2%7D%3D%5Csqrt%7B9%2B4%7D%3D%5Csqrt%7B13%7D)
![\dfrac{3}{\sqrt{13}}\sin x+\dfrac{2}{\sqrt{13}}\cos x=\dfrac{3}{\sqrt{13}}](https://tex.z-dn.net/?f=%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%5Csin+x%2B%5Cdfrac%7B2%7D%7B%5Csqrt%7B13%7D%7D%5Ccos+x%3D%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D)
Сделали это для того, чтобы теперь наш корень из суммы квадратов коэффициентов был равен единице. Проверим:
![\sqrt{\left(\dfrac{3}{\sqrt{13}}\right)^2+\left(\dfrac{2}{\sqrt{13}}\right)^2}=\sqrt{\dfrac{9+4}{13}}=\sqrt{\dfrac{13}{13}}=\sqrt{1}=1](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cleft%28%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%5Cright%29%5E2%2B%5Cleft%28%5Cdfrac%7B2%7D%7B%5Csqrt%7B13%7D%7D%5Cright%29%5E2%7D%3D%5Csqrt%7B%5Cdfrac%7B9%2B4%7D%7B13%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B13%7D%7B13%7D%7D%3D%5Csqrt%7B1%7D%3D1)
Так как это верное равенство, значит, числа
и
лежат на единичной окружности. Соответственно, существует такой угол
, что, например,
и
. Отсюда возьмём
.
![\sin\varphi\sin x+\cos\varphi\cos x=\dfrac{3}{\sqrt{13}}\medskip\\\cos\left(x-\varphi\right)=\dfrac{3}{\sqrt{13}}\medskip\\x-\varphi=\pm\arccos\dfrac{3}{\sqrt{13}}+2\pi n,\,n\in\mathbb{Z}\medskip\\x=\varphi\pm\arccos\dfrac{3}{\sqrt{13}}+2\pi n,\,n\in\mathbb{Z}\medskip\\x=\arcsin\dfrac{3}{\sqrt{13}}\pm\arccos\dfrac{3}{\sqrt{13}}+2\pi n,\,n\in\mathbb{Z}](https://tex.z-dn.net/?f=%5Csin%5Cvarphi%5Csin+x%2B%5Ccos%5Cvarphi%5Ccos+x%3D%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%5Cmedskip%5C%5C%5Ccos%5Cleft%28x-%5Cvarphi%5Cright%29%3D%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%5Cmedskip%5C%5Cx-%5Cvarphi%3D%5Cpm%5Carccos%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D%5Cmedskip%5C%5Cx%3D%5Cvarphi%5Cpm%5Carccos%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D%5Cmedskip%5C%5Cx%3D%5Carcsin%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%5Cpm%5Carccos%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D)
Можно наш ответ "разорвать" и привести к более благородному виду:
![\left[\begin{gathered}x=\arcsin\dfrac{3}{\sqrt{13}}+\arccos\dfrac{3}{\sqrt{13}}+2\pi n,\,n\in\mathbb{Z}\\x=\arcsin\dfrac{3}{\sqrt{13}}-\arccos\dfrac{3}{\sqrt{13}}+2\pi n,\,n\in\mathbb{Z}\end{gathered}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Bgathered%7Dx%3D%5Carcsin%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%2B%5Carccos%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D%5C%5Cx%3D%5Carcsin%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D-%5Carccos%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D%5Cend%7Bgathered%7D)
![\left[\begin{gathered}x=\dfrac{\pi}{2}+2\pi n,\,n\in\mathbb{Z}\\x=2\arcsin\dfrac{3}{\sqrt{13}}-\dfrac{\pi}{2}+2\pi n,\,n\in\mathbb{Z}\end{gathered}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Bgathered%7Dx%3D%5Cdfrac%7B%5Cpi%7D%7B2%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D%5C%5Cx%3D2%5Carcsin%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D-%5Cdfrac%7B%5Cpi%7D%7B2%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D%5Cend%7Bgathered%7D)
Ответ. ![x=\dfrac{\pi}{2}+2\pi n,\,n\in\mathbb{Z}\,;~x=2\arcsin\dfrac{3}{\sqrt{13}}-\dfrac{\pi}{2}+2\pi n,\,n\in\mathbb{Z}](https://tex.z-dn.net/?f=x%3D%5Cdfrac%7B%5Cpi%7D%7B2%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D%5C%2C%3B~x%3D2%5Carcsin%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D-%5Cdfrac%7B%5Cpi%7D%7B2%7D%2B2%5Cpi+n%2C%5C%2Cn%5Cin%5Cmathbb%7BZ%7D)