Question

Resolva a equação 2sen²(x) + cos²(x) =3/2 na variável x E [0, 2π].
por favor.

Answer 1

$\displaystyle \sf 2sen^2x+cos^2x=\frac{3}{2} \ \ ; \ \ x\in[0,2\pi ]$

Sabemos pela relação fundamental da trigonometria que :

$\sf sen^2x+cos^2x=1$

Então vamos fazer a relação fundamental aparecer, assim :

$\displaystyle \sf 2sen^2x+cos^2x=\frac{3}{2} \\\\ sen^2x+sen^2x+cos^2x= \frac{3}{2} \\\\\\ sen^2x+\underbrace{\sf sen^2x+cos^2x}_{1} = \frac{3}{2} \\\\\\ sen^2x+1=\frac{3}{2} \\\\\\ sen^2x=\frac{3}{2}- 1 \\\\\\ sen^2x=\frac{1}{2} \to sen\ x=\pm\frac{1}{\sqrt{2}} \\\\ racionalizando:\\\\ sen\ x= \pm\frac{\sqrt{2}}{2}$

$\sf \displaystyle sen \ x=\frac{\sqrt{2}}{2}\to x= \frac{\pi }{4}\ \ ou \ \ x = \frac{3\pi}{4} \\\\\\\ sen\ x =\frac{-\sqrt{2}}{2} \to x = \frac{5\pi }{4} \ \ ou \ x=\frac{7\pi}{4}$

Portanto o conjunto são é :

$\displaystyle \boxed{\sf S=\left\{\frac{\pi}{4}\ , \ \frac{3\pi }{4}\ , \ \frac{5\pi}{4}\ , \ \frac{7\pi}{4}\ \right\} }\checkmark$