Question

Помогите решить уравнение 9cosx......

Answer 1

$\displaysyle 9^{cosx}=9^{sinx}*3^{\frac{2}{cosx}}\\\\\displaysyle 3^{2cosx}=3^{2sinx}*3^{\frac{2}{cosx}}\\\\3^{2cosx}=3^{2sinx+\frac{2}{cosx}}\\\\2cosx=2sinx+\frac{2}{cosx}\,\,\,\,\,|*cosx \neq 0\\\\cos^2x=sinx*cosx+1\\\\cos^2x-sinx*cosx-1=0\\\\cos^2x-sinx*cosx-sin^2x-cos^2x=0\\\\sin^2x+sinx*cosx=0\\\\sinx(sinx+cosx)=0\\\\\\ \left[\begin{array}{ccc}sinx=0\\sinx+cosx=0\,\,|:cosx \neq 0\end{array}\right=\ \textgreater \ \left[\begin{array}{ccc}x=\pi n;\,\,\,n\in Z\\tgx+1=0\end{array}\right=\ \textgreater$

$\left[\begin{array}{ccc}x=\pi n;\,\,\,n\in Z\\tgx=-1\end{array}\right=\ \textgreater \ \boxed{\left[\begin{array}{ccc}x=\pi n;\,\,\,n\in Z\\x=-\frac{\pi}4+\pi n;\,\,\,n\in Z\end{array}\right}$