Question

Trigonometria FUVEST ! (Segunda Fase...)

Determine todos os valores de x pertencentes ao intervalo [0; 2π] que satisfazem à equação :

cos²(2*x) = 1/2 - sen²(x)

Answer 1

$Dada \ a \ express\tilde{a}o \ , \\\\ cos^2(2x) \ = \ \frac{1}{2} - sen^2(x) \ \ \ \ \ \ \ \ \ \ \ \ , \ \ x \in \ \Big[ \ 0 \ , \ 2\pi \ \Big] \\ \\ 2. \Big[cos(2x) \Big]^2 \ = \ 1 \ - 2.sen^2(x) \ \ \ \ \ \ \ \ \ \ (i) \\ \\ \\Irei \ colocar \ as \ demais \ inc\acute{o}gnitas \ em \ fun\c{c}\tilde{a}o \ de \ sen(x) \ . \ Para \\ isso \ utilizaremos \ : \\\\ \boxed{\boxed{cos(2x) \ = \ 1 \ - \ 2.sen^2(x) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (ii) \ \ \ \ \ \ \ }}$

$Aplicando \ a \ express\tilde{a}o \ (ii) \ em \ (i) \ , \ \\\\ 2 \ . \ \Big[ \ 1 \ - \ 2.sen^2(x) \ \Big]^2 \ = \ 1 \ - \ 2.sen^2(x) \\ \\ 2 \ . \ \Big[ \ 1 \ - \ 4.sen^2(x) \ + \ 4.sen^4(x) \ \Big] \ = \ 1 \ - \ 2.sen^2(x) \\ \\ 2 \ - \ 8.sen^2(x) \ + \ 8.sen^4(x) \ -\ 1 \ + \ 2.sen^2(x) \ = \ 0 \\ \\ 1 \ - \ 6.sen^2(x) \ + \ 8.sen^4(x) \ = \ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (iii)$

$Utilizando \ a \ propriedade \ de \ mudan\c{c}a \ de \ inc\acute{o}nita \ , \\ \\ \boxed{\boxed{ \ \ \ sen^2(x) \ = \ y \ \ \ \ \ \ (iiii) \ \ \ \ }} \\ \\ \\ Aplicando \ (iiii) \ em \ (iiii) \\\\ 1 \ - \ 6.y \ + 8y^2 \ = \ 0 \\ \\ Pelas \ relac{c}\tilde{o}es \ de \ Girard \ , \ temos \ que \ as \ ra\acute{i}zes \ da \ equac{c}\tilde{a}o \ s\tilde{a}o \ : \\ \\ y' \ = \ \frac{1}{2} \\ \\ y'' \ = \ \frac{1}{4} \\ \\ Retomando \ a \ (iiii) \ ,$

$sen^2(x') \ = \ y' \\ \\ sen^2(x') \ = \ \frac{1}{2} \\ \\ sen(x') \ = \ ^+ _- \ \frac{ \sqrt{2} }{2} \\ \\ Logo \ , \ x' \ representam \ todos \ m\acute{u}ltiplos \ de \ \frac{\pi}{4} \ no \ intervalo \\ considerado \ , \\ \\ S' \ = \ \begin{Bmatrix} \frac{\pi}{4} & , & \frac{3\pi}{4} & , & \frac{5\pi}{4} & , & \frac{7\pi}{4} \end{Bmatrix}$

$Retomando \ a \ (iiii) \ , \\ \\ sen^2(x'') \ = \ y'' \\ \\ sen^2(x'') \ = \ \frac{1}{4} \\ \\ sen(x'') \ = \ ^+ _- \ \frac{1}{2} \\ \\ \\ Assim \ , \ temos \ que \ x'' \ representam \ todos \ os \ m\acute{u}ltiplos \ de \ \frac{\pi}{6} \ no \\ intervalo \ considerado \ . \ Logo \ , \\ \\ \\ S'' \ = \ \begin{Bmatrix} \frac{\pi}{6} & , & \frac{5\pi}{6} & . & \frac{7\pi}{6} & . & \frac{11\pi}{6} \end{Bmatrix}$

$A \ solu\c{c}\tilde{a}o \ do \ sistema \ \acute{e} \ dada \ pela \ uni}\tilde{a}o \ das \ solu\c{c}\tilde{o}es \ S' \ e \ S'' \ . \\ Sendo \ assim, \\ \\ \\ S \ = \ S' \ \cup \ S'' \\ \\ S \ = \ \begin{Bmatrix} \frac{\pi}{6} \ , \ \frac{\pi}{4} \ , \ \frac{3\pi}{4} \ , \ \frac{5\pi}{6} \ , \ \frac{7\pi}{6} \ , \ \frac{5\pi}{4} \ , \ \frac{7\pi}{4} \ , \ \frac{11\pi}{6} \end{Bmatrix}$