Odpowiedź:
[tex]sinx=\frac{56\sqrt{2} }{81}[/tex]
Szczegółowe wyjaśnienie:
[tex]sin(\frac{x}{4} )=\frac{1}{3} \\x \in (0^{\circ},90^{\circ})[/tex]
Najpierw obliczmy [tex]cos(\frac{x}{4})[/tex] :
[tex]cos(\frac{x}{4}) =\sqrt{1-sin^2(\frac{x}{4}) } =\sqrt{1-\frac{1}{9} } =\frac{2\sqrt{2} }{3}[/tex]
Zauważmy, że:
[tex]sin(x)=sin(4 \cdot \frac{x}{4} )=sin(2 \cdot 2 \cdot \frac{x}{4} )=2sin( 2 \cdot \frac{x}{4})cos( 2 \cdot \frac{x}{4})=2 \cdot 2sin(\frac{x}{4} )cos(\frac{x}{4}) \cdot (1-2sin^{2}(\frac{x}{4} ))=4sin(\frac{x}{4} )cos(\frac{x}{4}) \cdot (1-2sin^{2}(\frac{x}{4} ))[/tex]
Podstawiamy wartości:
[tex]4sin(\frac{x}{4} )cos(\frac{x}{4}) \cdot (1-2sin^{2}(\frac{x}{4} ))=4 \cdot \frac{1}{3} \cdot \frac{2\sqrt{2} }{3} (1-2 \cdot \frac{1}{9} )=\frac{8\sqrt{2} }{9} \cdot \frac{7}{9} =\frac{56\sqrt{2} }{81}[/tex]
