Trigonometrie plana si sferica

Trigonometrie plana si sferica

Relatii intre laturi si unghiuri intr-un triunghi oarecare

Fie $$\triangle ABC$$ oarecare. Avem:

1. Unghiurile sunt notate cu $$A$$, $$B$$ si $$C$$ si masura lor este cuprinsa intre $$0^0$$ si $$180^0$$ (in radiani intre $$0$$ si $$\pi$$):

@dA+B+C=180^0@d

2. Laturile se noteaza cu $$a=BC,\ b=CA,\ c=AB$$ si verifica inegalitatile:    \begin{eqnarray}
    a<b+c,&b<c+a,&c<a+b\\
    a>|b-c|,&b>|c-a|,&c>|a-b|
    \end{eqnarray}

3. Teorema sinusurilor:

@d\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=\frac{abc}{2S}=2R@d

unde $$S$$ este aria triunghiului si $$R$$ este raza cercului circumscris triunghiului

4. Teorema proiectiilor:

\begin{eqnarray*}
&a=c\cos B+b\cos C\\
&b=a\cos C+c\cos A\\
&c=b\cos A+a\cos B
\end{eqnarray*}

5. Teorema lui Pitagora generalizata:

\begin{eqnarray*}
&a^2=b^2+c^2-2bc\cos A\\
&b^2=a^2+c^2-2ac\cos B\\
&c^2=a^2+b^2-2ab\cos C
\end{eqnarray*}

6. Teorema cosinusului:

\begin{eqnarray*}
&\cos A=\displaystyle\frac{b^2+c^2-a^2}{2bc}\\
&\cos B=\displaystyle\frac{a^2+c^2-b^2}{2ac}\\
&\cos C=\displaystyle\frac{a^2+b^2-c^2}{2ab}
\end{eqnarray*}

7. Teorema tangentei:
@d\frac{a-b}{a+b}=\frac{\operatorname{tg}\frac{A-B}{2}}{\operatorname{tg}\frac{A+B}{2}},\
\frac{b-c}{b+c}=\frac{\operatorname{tg}\frac{B-C}{2}}{\operatorname{tg}\frac{B+C}{2}},\ \frac{c-a}{c+a}=
\frac{\operatorname{tg}\frac{C-A}{2}}{\operatorname{tg}\frac{C+A}{2}}@d

8. Formulele lui Mollweide:

@d\frac{a+b}{c}=\frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}},\ \frac{b+c}{a}=\frac{\cos\frac{B-C}{2}}{\sin\frac{A}{2}},\ \frac{c+a}{b}=\frac{\cos\frac{C-A}{2}}{\sin\frac{B}{2}}@d

@d\frac{a-b}{c}=\frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}},\ \frac{b-c}{a}=\frac{\sin\frac{B-C}{2}}{\cos\frac{A}{2}},\ \frac{c-a}{b}=\frac{\sin\frac{C-A}{2}}{\cos\frac{B}{2}}@d

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