Trigonometrie plana si sferica

Trigonometrie plana si sferica

Formule pentru diverse elemente ale unui triunghi

1. Aria triunghiului

\begin{eqnarray*}S&=&\frac{1}{2}bc\sin A=\frac{1}{2}ac\sin B=\frac{1}{2}ab\sin C\\
&=&\sqrt{p(p-a)(p-b)(p-c)}\hspace{1cm}\text{(formula lui Heron)}\\
&=&\frac{a^2\sin B\sin C}{2\sin A}=\frac{b^2\sin A\sin C}{2\sin B}=\frac{c^2\sin A\sin B}{2\sin C}
\end{eqnarray*}

2. Raza cercului circumscris triunghiului

@d2R=\frac{abc}{2S}\Rightarrow R=\frac{abc}{4\sqrt{p(p-a)(p-b)(p-c)}}@d

3. Raza cercului inscris in triunghi

@dS=\frac{1}{2}ar+\frac{1}{2}br+\frac{1}{2}cr=\frac{a+b+c}{2}\cdot r= p\cdot r\Rightarrow r=\frac{S}{p}@d

4. Inaltimile triunghiului

\begin{eqnarray*}
&h_a=c\sin B=b\sin C=2R\sin B\sin C\\
&h_b=c\sin A=a\sin C=2R\sin A\sin C\\
&h_c=a\sin B=b\sin A=2R\sin A\sin B
\end{eqnarray*}

5. Bisectoarele triunghiului

\begin{eqnarray*}
&b_a=\displaystyle\frac{b\sin C}{\cos\frac{B-C}{2}}=\frac{c\sin B}{\cos\frac{B-C}{2}}\\
&b_b=\displaystyle\frac{c\sin A}{\cos\frac{C-A}{2}}=\frac{a\sin C}{\cos\frac{C-A}{2}}\\
&b_c=\displaystyle\frac{a\sin B}{\cos\frac{A-B}{2}}=\frac{b\sin A}{\cos\frac{A-B}{2}}
\end{eqnarray*}

6. Medianele triunghiului

\begin{eqnarray*}
&m_a^2=\frac{2(b^2+c^2)-a^2}{4}\\
&m_b^2=\frac{2(a^2+c^2)-b^2}{4}\\
&m_c^2=\frac{2(a^2+b^2)-c^2}{4}
\end{eqnarray*}

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