Синусна теорема. Примена на сложенијим примерима.
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Пр.1) Одредити остале елементе троугла ако су дати страница $a = 2\sqrt 2 $ и два његова угла $\alpha = {45^ \circ }$ и $\beta = {120^ \circ }$.
Пр.2) Дате су две странице троугла $a = \frac{{10\sqrt 3 }}{3},b = 10$ и угао наспрам једне од њих $\alpha = {30^ \circ }$. Одредти остале елементе тог троугла.
Пр.1 \[\begin{gathered}
\alpha + \beta + \gamma = {180^\circ } \hfill \\
{45^\circ } + {120^\circ } + \gamma = {180^\circ } \hfill \\
\gamma = {180^\circ } - \left( {{{45}^\circ } + {{120}^\circ }} \right) \hfill \\
\gamma = {15^\circ } \hfill \\
\end{gathered} \]
\[\begin{gathered}
\frac{a}{{\sin \alpha }} = \frac{b}{{\sin \beta }} \hfill \\
\sin \alpha = \sin {45^\circ } = \frac{{\sqrt 2 }}{2} \hfill \\
\sin \beta = \sin {120^\circ } = \frac{{\sqrt 3 }}{2} \hfill \\
\frac{{2\sqrt 2 }}{{\frac{{\sqrt 2 }}{2}}} = \frac{b}{{\frac{{\sqrt 3 }}{2}}} \hfill \\
2b = 4\sqrt 3 \hfill \\
b = 2\sqrt 3 \hfill \\
\end{gathered} \]
\[\begin{gathered}
\frac{a}{{\sin \alpha }} = \frac{c}{{\sin \gamma }} \hfill \\
\sin \gamma = \sin {15^\circ } = \sin \frac{{{{30}^\circ }}}{2} = \sqrt {\frac{{1 - \cos {{30}^\circ }}}{2}} = \sqrt {\frac{{1 - \frac{{\sqrt 3 }}{2}}}{2}} = \hfill \\
= \sqrt {\frac{{2 - \sqrt 3 }}{4}} = \frac{{\sqrt {2 - \sqrt 3 } }}{2} \hfill \\
\frac{{2\sqrt 2 }}{{\frac{{\sqrt 2 }}{2}}} = \frac{c}{{\frac{{\sqrt {2 - \sqrt 3 } }}{2}}} \hfill \\
4 = \frac{c}{{\frac{{\sqrt {2 - \sqrt 3 } }}{2}}} \hfill \\
4 = \frac{{2c}}{{\sqrt {2 - \sqrt 3 } }} \hfill \\
2c = 4\sqrt {2 - \sqrt 3 } \hfill \\
c = 2\sqrt {2 - \sqrt 3 } \hfill \\
\end{gathered} \]
Пр.2 Знамо да је $b > a$ онда $\beta > \alpha $.
\[\begin{gathered}
\frac{a}{{\sin \alpha }} = \frac{b}{{\sin \beta }} \hfill \\
b\sin \alpha = a\sin \beta \hfill \\
\sin \beta = \frac{b}{a}\sin \alpha \hfill \\
\sin \beta = \frac{{10}}{{\frac{{10\sqrt 3 }}{3}}} \cdot \frac{1}{2} \hfill \\
\sin \beta = \frac{3}{{\sqrt 3 }} \cdot \frac{1}{2} \hfill \\
\sin \beta = \frac{{\sqrt 3 }}{2} \hfill \\
1)\beta = {60^\circ } \hfill \\
\gamma = {180^\circ } - \left( {\alpha + \beta } \right) \hfill \\
\gamma = {90^\circ } \hfill \\
{c^2} = {a^2} + {b^2} \hfill \\
{c^2} = {\left( {\frac{{10\sqrt 3 }}{3}} \right)^2} + {10^2} \hfill \\
{c^2} = \frac{{100 \cdot 3}}{9} + 100 \hfill \\
{c^2} = \frac{{400}}{3} \hfill \\
c = \frac{{20}}{{\sqrt 3 }} \cdot \frac{{\sqrt 3 }}{{\sqrt 3 }} \hfill \\
c = \frac{{20\sqrt 3 }}{3} \hfill \\
2)\beta = {120^\circ } \hfill \\
\gamma = {180^\circ } - \left( {\alpha + \beta } \right) \hfill \\
\gamma = {30^\circ } \hfill \\
\alpha = \gamma \hfill \\
a = c \hfill \\
c = \frac{{10\sqrt 3 }}{3} \hfill \\
\end{gathered} \]