Trigonometrisk likning [VGS]

[Drezky]

Løs likninga [tex]4\left ( 16^{\sin ^2 x} \right )=2^{6 \sin x}[/tex] for [tex]x\in \left [ 0,2\pi \right ][/tex]

[DennisChristensen]

Løs likninga [tex]4\left ( 16^{\sin ^2 x} \right )=2^{6 \sin x}[/tex] for [tex]x\in \left [ 0,2\pi \right ][/tex]

$\displaystyle\begin{align*} 2^2 \cdot (2^4)^{\sin ^2 x} & = 2^{6\sin x} \\
2^{2 + 4\sin ^2 x} & = 2^{6\sin x} \\
2 + 4\sin ^2 x & = 6\sin x \\
1 + 2\sin ^2 x & = 3\sin x \end{align*}$

$ABC$-formelen gir $\displaystyle \sin x = \frac{3 \pm \sqrt{3^2 - 4\cdot 2\cdot 1}}{2\cdot 2} = \frac{3 \pm \sqrt{9 - 8}}{4} = \frac{3}{4} \pm \frac{1}{4}$.

Vi har at $x \in [0,2\pi)$, så

$\displaystyle \sin x = \frac{3}{4} - \frac{1}{4} = \frac{1}{2}$ gir løsningene $\displaystyle x = \frac{\pi}{6}$ og $\displaystyle x = \frac{5\pi}{6}$.

$\displaystyle \sin x = \frac{3}{4} + \frac{1}{4} = 1$ gir løsningen $\displaystyle x = \frac{\pi}{2}$.