Tożsamość trygonometryczna

[mati24568]

Oblicz \(\displaystyle{ \frac{\sin\alpha}{\sin^3\alpha+3\cos^3\alpha}}\) wiedząc, że \(\displaystyle{ \tg\alpha=2}\)

[anna_]

Policz \(\displaystyle{ sin\alpha}\) i \(\displaystyle{ cos\alpha}\) z
\(\displaystyle{ \begin{cases} \frac{sin\alpha}{cos\alpha} =2 \\ sin^2\alpha+cos^2\alpha=1 \end{cases}}\)

[ares41]

\(\displaystyle{ \frac{\sin\alpha}{\sin^3\alpha+3\cos^3\alpha}= \frac{1}{ \frac{\sin^3\alpha+3\cos^3\alpha}{\sin\alpha} }= \frac{1}{\sin^2{\alpha}+3\cos^2{\alpha} \cdot \ctg{\alpha}}= \frac{1}{\sin^2{\alpha}+3\cos^2{\alpha} \cdot \frac{1}{\tg{\alpha}} }=\frac{1}{\sin^2{\alpha}+3\cos^2{\alpha} \cdot \frac{1}{2} }=\frac{1}{\sin^2{\alpha}+\cos^2{\alpha}+0,5\cos^2{\alpha}}=\ldots}\)
cosinusa wyliczysz ze wzoru na tangensa i jedynki trygonometrycznej.

[mati24568]

Dzięki