begin{gathered}sin2x+5(sinx+cosx)=0t=sinx+cosx\; , t^2=sin^2x+cos^2x+2sinx\cdot cosx=1+sin2x,sin2x=t^2-1t^2+5t-1=0D=25+4=29\; ,\; t_{1,2}= \frac{-5\pm \sqrt{29}}{2} t_1\approx-5,19\; \; ;\; \; t_2\approx0,19a)\quad sinx+cosx= \frac{-5-\sqrt{29}}{2} \, \left |:\sqrt2\end{gathered}
begin{gathered}sin2x+5(sinx+cosx)=0t=sinx+cosx\; , t^2=sin^2x+cos^2x+2sinx\cdot cosx=1+sin2x,sin2x=t^2-1t^2+5t-1=0D=25+4=29\; ,\; t_{1,2}= \frac{-5\pm \sqrt{29}}{2} t_1\approx-5,19\; \; ;\; \; t_2\approx0,19a)\quad sinx+cosx= \frac{-5-\sqrt{29}}{2} \, \left |:\sqrt2\end{gathered}