Question

Sa se rezolve in intervalul [0,2pi] ecuatia sin^4 x +cos^4 x=1

Answer 1

$\displaystyle Avem~sin^2x+cos^2x=1,~si~prin~ridicare~la~patrat~obtinem: \\ \\ sin^4x+2sin^2xcos^2x+cos^4x=1. \\ \\ Dar~sin^4x+cos^4x=1~(conform~enuntului) \Rightarrow 2sin^2xcos^2x=0. \\ \\ Avem~doua~posibilitati:~sinx=0~sau~cosx=0. \\ \\ i)~sinx=0 \Rightarrow x \in \{0; \pi ; 2\pi \} \\ \\ Toate~aceste~valori~verifica~relatia~din~ipoteza. \\ \\ ii)~cosx=0 \Rightarrow x \in \left \{ \frac{\pi}{2}; \frac{3 \pi}{2}\right \}. \\ \\ Toate~aceste~valori~verifica~relatia~din~ipoteza.$

$\displaystyle Solutie:~x \in \left \{ 0; \frac{ \pi}{2}; \pi; \frac{3 \pi}{2}; 2 \pi \right \}.$