Dla jakich wartosci parametru m (trygonometria)?

[julietta_m_18]

Dla jakich wartości parametru m liczby \(\displaystyle{ sin\alpha}\) i \(\displaystyle{ cos\alpha}\) są pierwiastkami równania \(\displaystyle{ x^{2}+mx-\frac{1}{4}=0}\)

[bullay]

\(\displaystyle{ sin^2\alpha+msin\alpha-\frac{1}{4}=0}\) i \(\displaystyle{ cos^2\alpha+mcos\alpha-\frac{1}{4}=0}\)
\(\displaystyle{ sin^2\alpha+msin\alpha=cos^2\alpha+mcos\alpha}\)
\(\displaystyle{ sin^2\alpha-cos^2\alpha=m(cos\alpha-sin\alpha)}\)
\(\displaystyle{ \frac{(sin\alpha+cos\alpha)(sin\alpha-cos\alpha)}{cos\alpha-sin\alpha}=m}\)
\(\displaystyle{ m=-(sin\alpha+cos\alpha)}\)

zal: \(\displaystyle{ sin\alpha cos\alpha}\)

[jacek_ns]

\(\displaystyle{ x_{1}=sin\alpha\\ x_{2}=cos\alpha}\)

\(\displaystyle{ sin^{2}\alpha+cos^{2}\alpha=1 \Rightarrow x_{1}^{2}+x_{2}^{2}=1}\)

\(\displaystyle{ \begin{cases} \Delta >0 \\ x_{1}^{2}+x_{2}^{2}=1 \end{cases}}\)

\(\displaystyle{ x_{1}^{2}+x_{2}^{2}=(x_{1}+x_{2})^{2}-2x_{1}x_{2}}\)