log-2-sin-x-1-cos-x-2-2pi-3-x-pi-3-

Question Number 131661 by benjo_mathlover last updated on 07/Feb/21

\log_{\sqrt{2} \sin x} (1 + \cos x) = 2

- \frac{2 π}{3} ⩽ x ⩽ \frac{π}{3}

Answered by liberty last updated on 07/Feb/21

{\begin{cases} \sqrt{2} \sin x > 0; x in I or II quadrant \\ \sqrt{2} \sin x \neq 1 \Rightarrow \sin x \neq \frac{1}{\sqrt{2}} \end{cases}

\Leftrightarrow 1 + \cos x = {(\sqrt{2} \sin x)}^{2}

\Leftrightarrow 2 \sin^{2} x - \cos x - 1 = 0

\Leftrightarrow 2 - 2 \cos^{2} x - \cos x - 1 = 0

\Leftrightarrow 2 \cos^{2} x + \cos x - 1 = 0

\Leftrightarrow (2 \cos x - 1) (\cos x + 1) = 0

{\begin{cases} \cos x = \frac{1}{2} \Rightarrow x = \frac{π}{3} \leftarrow only solution \\ \cos x = - 1 (reject) \end{cases}

Commented by liberty last updated on 07/Feb/21

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