X < 95
x ∈ ( - ∞; 95)
---------------------
Y = Sin3x - Cos3x
y ' = (Sin3x)' - (Cos3x)' = Сos3x * (3x)' + Sin3x * (3x)' = 3Cos3x + 3Sin3x =
= 3(Cos3x + Sin3x)
![3(Cos(3* \frac{3 \pi }{4}) +Sin(3* \frac{3 \pi }{4}))=3(Cos \frac{9 \pi }{4}+Sin \frac{9 \pi }{4})=3[Cos(2 \pi + \frac{ \pi }{4})](https://tex.z-dn.net/?f=3%28Cos%283%2A+%5Cfrac%7B3+%5Cpi+%7D%7B4%7D%29+%2BSin%283%2A+%5Cfrac%7B3+%5Cpi+%7D%7B4%7D%29%29%3D3%28Cos+%5Cfrac%7B9+%5Cpi+%7D%7B4%7D%2BSin+%5Cfrac%7B9+%5Cpi+%7D%7B4%7D%29%3D3%5BCos%282+%5Cpi+%2B+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%29+++)
![+Sin(2 \pi + \frac{ \pi }{4})]=3(Cos \frac{ \pi }{4} +Sin \frac{ \pi }{4})=3( \frac{ \sqrt{2} }{2} + \frac{ \sqrt{2} }{2})=3*2* \frac{ \sqrt{2} }{2}=3 \sqrt{2}](https://tex.z-dn.net/?f=%2BSin%282+%5Cpi+%2B+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%29%5D%3D3%28Cos+%5Cfrac%7B+%5Cpi+%7D%7B4%7D+%2BSin+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%29%3D3%28++%5Cfrac%7B+%5Csqrt%7B2%7D+%7D%7B2%7D+%2B+%5Cfrac%7B+%5Csqrt%7B2%7D+%7D%7B2%7D%29%3D3%2A2%2A+%5Cfrac%7B+%5Csqrt%7B2%7D+%7D%7B2%7D%3D3+%5Csqrt%7B2%7D+++++)
Ответ:
![x_1=2\pi n, n \in Z\\x_2=\frac{3\pi}{2}+2\pi k, k \in Z](https://tex.z-dn.net/?f=x_1%3D2%5Cpi%20n%2C%20n%20%5Cin%20Z%5C%5Cx_2%3D%5Cfrac%7B3%5Cpi%7D%7B2%7D%2B2%5Cpi%20k%2C%20k%20%5Cin%20Z)
Объяснение:
Преобразуем левую часть уравнения:
![cos(x)-sin(x)=\sqrt{1^2+1^2}*(\frac{1}{\sqrt{1^2+1^2}}cos(x)-\frac{1}{\sqrt{1^2+1^2}}sin(x))=\sqrt{2}(sin(\frac{\pi}{4})cos(x)-cos(\frac{\pi}{4})sin(x))=\sqrt{2}sin(\frac{\pi}{4}-x)](https://tex.z-dn.net/?f=cos%28x%29-sin%28x%29%3D%5Csqrt%7B1%5E2%2B1%5E2%7D%2A%28%5Cfrac%7B1%7D%7B%5Csqrt%7B1%5E2%2B1%5E2%7D%7Dcos%28x%29-%5Cfrac%7B1%7D%7B%5Csqrt%7B1%5E2%2B1%5E2%7D%7Dsin%28x%29%29%3D%5Csqrt%7B2%7D%28sin%28%5Cfrac%7B%5Cpi%7D%7B4%7D%29cos%28x%29-cos%28%5Cfrac%7B%5Cpi%7D%7B4%7D%29sin%28x%29%29%3D%5Csqrt%7B2%7Dsin%28%5Cfrac%7B%5Cpi%7D%7B4%7D-x%29)
Отсюда получим уравнение:
![\sqrt{2}sin(\frac{\pi}{4}-x)=1\\sin(\frac{\pi}{4}-x)=\frac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=%5Csqrt%7B2%7Dsin%28%5Cfrac%7B%5Cpi%7D%7B4%7D-x%29%3D1%5C%5Csin%28%5Cfrac%7B%5Cpi%7D%7B4%7D-x%29%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
Найдем общее решение уравнения.
![sin(\frac{\pi}{4}-x)=\frac{\sqrt{2}}{2}\\sin(x-\frac{\pi}{4})=-\frac{\sqrt{2}}{2}\\x-\frac{\pi}{4}=(-1)^narcsin(-\frac{\sqrt{2}}{2})+\pi n\\x=\frac{\pi}{4}+(-1)^{n+1}\frac{\pi}{4}+\pi n, n \in Z](https://tex.z-dn.net/?f=sin%28%5Cfrac%7B%5Cpi%7D%7B4%7D-x%29%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5C%5Csin%28x-%5Cfrac%7B%5Cpi%7D%7B4%7D%29%3D-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5C%5Cx-%5Cfrac%7B%5Cpi%7D%7B4%7D%3D%28-1%29%5Enarcsin%28-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29%2B%5Cpi%20n%5C%5Cx%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%2B%28-1%29%5E%7Bn%2B1%7D%5Cfrac%7B%5Cpi%7D%7B4%7D%2B%5Cpi%20n%2C%20n%20%5Cin%20Z)
Или же можно записать так:
![x_1=2\pi n, n \in Z\\x_2=\frac{3\pi}{2}+2\pi k, k \in Z](https://tex.z-dn.net/?f=x_1%3D2%5Cpi%20n%2C%20n%20%5Cin%20Z%5C%5Cx_2%3D%5Cfrac%7B3%5Cpi%7D%7B2%7D%2B2%5Cpi%20k%2C%20k%20%5Cin%20Z)
3а+5b=0
5b=-3a
b=-06a
4a+9*(-06a)/5a-(-06a)=4a-5,4a/5,6a=-1,4a/5,6a=-0,25