Question

sin x+ cos x= -1 , x aparține [0, 2 pi)

Answer 1

$sin \ x+ cos \ x = -1 \ \ ; \ \ x \in [0;2 \pi] \\\\ \hbox{Ridicam la patrat ecuatia si astfel obtinem: } \\\\\\ (sin \ x+cos \ x)^2= (-1)^2 \\\\ \underbrace{sin^2 x +cos^2 x}_{1}+ 2* sin \ x *cos \ x=1 \\\\\\ \not{1}+ 2* sin \ x *cos \ x=\not{1} \\\\\\ 2* sin \ x *cos \ x=0 \\\\\\\hbox{Luam pe cazuri:} \\\\ I) sin \ x=0 \\\\ x \in \{(-1)^k * \underbrace{arcsin(0)}_{0} + k\pi / k \in Z\} \\\\ x \in \{ k\pi/k \in Z\} \\\\ Pt \ k =0 \to x=0 \ \ 'A' \\ k=1 \to x=\pi \ \ 'A' \\ k=2 \to x=2\pi \ \ 'A'$

$k=3 \to x=3\pi \ \ 'F' \\\\\\S_1= \{0;\pi;2\pi\} \\\\\\ II) cos \ x=0 \\\\\\ x\in \{ \pm \underbrace{arcos(0)}_{\frac{\pi}{2}}+ 2k\pi/ k \in Z \} \\\\ x \in \{ \pm \frac{\pi}{2} + 2k\pi/k \in Z\} \\\\\\ Pt \ k=0 \to x= \pm \frac{\pi}{2} \ \ \ \left \{ {{x=-\frac{\pi}{2}} \ \ 'F'\atop {x=\frac{\pi}{2}} \ \ 'A'}} \right. \\\\ k=1\to x=\pm \frac{\pi}{2}} +2\pi \ \ \ \left \{ {{x=-\frac{\pi}{2}} +2\pi \to \frac{3\pi}{2}} \ \ 'A'} \atop {x=\frac{\pi}{2}} +2\pi \to \frac{5\pi}{2}} \ \ 'F'}} \right.$

$\\\\ k=2\to x= \pm \frac{\pi}{2}} + 4\pi \ \ \ \left \{ {{x=-\frac{\pi}{2}} +4\pi \to \frac{7\pi}{2}} \ \ 'F'} \atop {x=\frac{\pi}{2}} +4\pi\to \frac{9\pi}{2}} \ \ 'F' }} \right. \\\\\\ S_2=\{\frac{\pi}{2}} ;\frac{3\pi}{2}} \} \\\\\\ \underline{S=S_1 \cup S_2 } \\\\ \boxed{\boxed{S=\{0;\frac{\pi}{2} ; \pi; \frac{3\pi}{2} ; 2\pi \}}}$

$\hbox{Verificare: \ \ sin \ x + cos \ x= -1} \\\\ Pt \ x\to 0 \ \ \ \underbrace{sin \ 0}_{0}+ \underbrace{cos \ 0}_{1}= -1 \ \ \ 'F' \\\\\\ x\to \frac{\pi}{2} \ \ \ \underbrace{sin \frac{\pi}{2}}_{1}+\underbrace{cos \frac{\pi}{2}}_{0} = -1 \ \ \ 'F' \\\\\\ x \to \pi \ \ \ \underbrace{sin \pi}_{0}+ \underbrace{cos \pi}_{-1}=-1 \ \ \ 'A' \\\\\\ x \to \frac{3\pi}{2} \ \ \ \underbrace{sin \frac{3\pi}{2}}_{-1} + \underbrace{cos \frac{3\pi}{2} }_{0}= -1 \ \ \ 'A'$

$x \to 2\pi \ \ \ \underbrace{sin \ 2\pi}_{0}+ \underbrace{cos \ 2\pi}_{1}= -1 \ \ \ 'F' \\\\ \hbox{Astfel ramanem doar cu solutiile: } \\\\ \boxed{\underline{S_{f} \in \{\pi; \frac{3\pi}{2} \}}}$