Question

sinx+cosx=2.5+5sinx*cosx

Answer 1

$sinx+cosx=2,5+5sinx\cdot cosx\\\\Zamena:\; \; t=sinx+cosx\\\\t^2=(sinx+cosx)^2=\underbrace {sin^2x+cos^2x}_{1}+2sinx\cdot cosx=1+2sinx\cdot cosx\; \to \\\\2sinx\cdot cosx=t^2-1\; ,\quad sinx\cdot cosx=\frac{t^2-1}{2}\\\\t=2,5+5\cdot \frac{t^2-1}{2}\; \; ,\; \; \; t=\frac{5}{2}+\frac{5(t^2-1)}{2}\; \; ,\; \; 2t=5+5t^2-5\; ,\\\\5t^2-2t=0\; \; ,\; \; \; t\, (5t-2)=0\; \; \to \; \; t=0\; \; ili\; \; 5t-2=0\\\\a)\; \; t=0\; \; ,\; \; \; sinx+cosx=0\, |:cosx\ne 0\; ,\\\\tgx+1=0\; \; ,\; \; tgx=-1\; ,\\\\x=-\frac{\pi}{4}+\pi n\; ,\; n\in Z$

$b)\; \; 5t-2=0\; \; ,\; \; t=\frac{2}{5}\; \; ,\; \; sinx+cosx=\frac{2}{5} \; ,\\\\sinx+sin(\frac{\pi}{2}-x)=\frac{2}{5}\\\\2sin\frac{x+(\frac{\pi}{2}+x)}{2}\cdot cos\frac{x-(\frac{\pi}{2}-x)}{2}=\frac{2}{5}\\\\2sin(x+\frac{\pi}{4})\cdot cos(-\frac{\pi}{4})=\frac{2}{5}\\\\\sqrt2\cdot sin(x+\frac{\pi}{4})=\frac{2}{5}\\\\sin(x+\frac{\pi }{4})=\frac{\sqrt2}{5}\\\\x+\frac{\pi}{4}=(-1)^{n}\cdot arcsin\frac{\sqrt2}{5}+\pi n\; ,\; n\in Z\\\\x=-\frac{\pi }{4}+(-1)^{n}\cdot arcsin\frac{\sqrt2}{5}+\pi n\; ,\; n\in Z$

$Otvet:\; \; x=-\frac{\pi}{4}+\pi n\; ,\; \; x=-\frac{\pi }{4}+(-1)^{n}\cdot arcsin\frac{\sqrt2}{5}+\pi n\; ,\; n\in Z\; .$