Question

Dê a solução geral da equação sen²x + cos³x = 7/8
Tem que ser bom para resolver essa.

Answer 1

Resposta:

Se $\displaystyle \mathtt{0 \leq x \leq 2\pi}$, então $\\ \displaystyle \boxed{\boxed{\mathtt{S = \left\{ \frac{\pi}{5}, \frac{\pi}{3}, \frac{3\pi}{5}, \frac{7\pi}{5}, \frac{9\pi}{5}, \frac{5\pi}{3} \right\}}}}$

Explicação passo a passo:

Inicialmente, temos:

$\\ \displaystyle \mathtt{\sin^2 x + \cos^3 x = \frac{7}{8}} \\\\ \mathtt{(1 - \cos^2 x) + \cos^3 x = \frac{7}{8}} \\\\ \mathtt{\cos^3 x - \cos^2 x = - \frac{1}{8}} \\\\ \mathtt{\cos^2 x \cdot \left ( \cos x - 1 \right ) = \frac{1}{4} \cdot \frac{- 1}{2}}$

Posto isto, temos que:

$\\ \displaystyle \mathtt{\cos x - 1 = - \frac{1}{2}} \\\\ \mathtt{\cos x = \frac{1}{2}} \\\\ \boxed{\mathtt{S_1 = \left\{ x \in \mathbb{R} / x = \frac{\pi}{3} + 2k\pi \ ou \ x = \frac{5\pi}{3} + 2k\pi \right\}}, \mathtt{\forall \ k \in \mathbb{Z}}}$

Tomando $\displaystyle \mathtt{\cos x = \lambda}$, teremos $\displaystyle \mathtt{\lambda^3 - \lambda^2 + \frac{1}{8} = 0}$, onde $\displaystyle \mathtt{\lambda - \frac{1}{2}}$ divide a equação. Efetuando a divisão tiramos que:

$\displaystyle \mathtt{8\lambda^3 - 8\lambda^2 + 1 = (\lambda - \frac{1}{2}) \cdot (8\lambda^2 - 4\lambda - 2)}$

Resolvendo o segundo fator, encontramos $\displaystyle \boxed{\mathtt{\lambda = \frac{1 + \sqrt{5}}{4}}}$ e $\displaystyle \boxed{\mathtt{\lambda = \frac{1 - \sqrt{5}}{4}}}$.

Com efeito,

$\displaystyle \left\{\begin{matrix} \mathtt{\cos x = \frac{1 + \sqrt{5}}{4} \Rightarrow x = \cos^{- 1} \left ( \frac{1 + \sqrt{5}}{4} \right ) \Rightarrow \boxed{\mathtt{x = 36^o}}} & \\\\ \mathtt{\cos x = \frac{1 - \sqrt{5}}{4} \Rightarrow x = \cos^{- 1} \left ( \frac{1 - \sqrt{5}}{4} \right ) \Rightarrow \boxed{\mathtt{x = 108^o}}} & \\ \end{matrix}\right.$

Isto é,

$\boxed{\mathtt{S_2 = \left\{ x \in \mathbb{R} / x = \frac{\pi}{5} + 2k\pi \ ou \ x = \frac{3\pi}{5} + 2k\pi \right\}}, \mathtt{\forall \ k \in \mathbb{Z}}}$

Por fim,

$\displaystyle \left\{\begin{matrix} \mathtt{x = 360^o - 36^o \Rightarrow \boxed{\mathtt{x = 324^o}}} & \\ \mathtt{x = 360^o - 108^o \Rightarrow \boxed{\mathtt{x = 252^o}}} & \\ \end{matrix}\right.$

Ou seja,

$\boxed{\mathtt{S_3 = \left\{ x \in \mathbb{R} / x = \frac{9\pi}{5} + 2k\pi \ ou \ x = \frac{7\pi}{5} + 2k\pi \right\}}, \mathtt{\forall \ k \in \mathbb{Z}}}$