Question

a) cos(2x + pi /3)=1/2

 

b)1/4 <ou= sen(x) . cos(x)<ou=raiz de2 /2

Answer 1

Olá, Josyeshua.

$a)\ \cos(2x + \frac \pi 3)=\frac12 \Rightarrow \\\\ 2x + \frac \pi 3=\frac \pi 3+2k\pi, k\in\mathbb{Z} \Rightarrow \\\\ 2x =2k\pi, k\in\mathbb{Z} \Rightarrow \\\\ \boxed{x =k\pi, k\in\mathbb{Z} }$

$b)\ \frac14 \leq \sin x \cos x \leq \frac{\sqrt2} 2 \Rightarrow \\\\ 2\cdot\frac14 \leq 2\sin x \cos x \leq 2\cdot\frac{\sqrt2} 2 \Rightarrow\\\\ \frac12\leq \sin 2x \leq \sqrt2 \Rightarrow \\\\ \frac \pi 6 + 2k\pi \leq 2x \leq \arcsin\sqrt2 + 2k\pi\Rightarrow$

$\boxed{\frac{\pi}{12}+k\pi \leq x \leq \frac12 \arcsin\sqrt2 + k\pi,k\in\mathbb{Z}}$