Тригонометријске функције оштрог угла – примери 2

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Пр.3   Одреди вредност израза:

           $\frac{{tg60^\circ  - tg30^\circ }}{{\sin 30^\circ  - \cos 60^\circ }} = $

Пр.4  Израчунати остале тригонометријске функције оштрог угла $\alpha $

           ако је: $\sin \alpha \frac{3}{5}$   $0 < \alpha  < \frac{\pi }{2}$ , и ако је

                      $tg\alpha  = \frac{7}{{24}}$   $0 < \alpha  < \frac{\pi }{2}$

Пр.3   

\[\frac{{tg{{60}^ \circ } - tg{{30}^ \circ }}}{{sin{{30}^ \circ } - cos{{60}^ \circ }}} = \frac{{\sqrt 3  - \frac{{\sqrt 3 }}{3}}}{{\frac{1}{2} + \frac{1}{2}}} = \frac{{\frac{{3\sqrt 3  - \sqrt 3 }}{3}}}{1} = \frac{{2\sqrt 3 }}{3}\]

Пр.4 

$\sin \alpha \frac{3}{5}$   $0 < \alpha  < \frac{\pi }{2}$ ,

\[\begin{gathered}
{\sin ^2}\alpha + {\cos ^2}\alpha = 1 \hfill \\
{\left( {\frac{3}{5}} \right)^2} + {\cos ^2}\alpha = 1 \hfill \\
\frac{9}{{25}} + {\cos ^2}\alpha = 1 \hfill \\
{\cos ^2}\alpha = 1 - \frac{9}{{25}} \hfill \\
{\cos ^2}\alpha = \frac{{16}}{{25}} \hfill \\
\cos \alpha = \frac{4}{5} \hfill \\
\end{gathered} \]

\[\begin{gathered}
tg\alpha = \frac{{\sin \alpha }}{{\cos \alpha }} = \frac{{\frac{3}{5}}}{{\frac{4}{5}}} = \frac{3}{4} \hfill \\
ctg\alpha = \frac{1}{{tg\alpha }} = \frac{4}{3} \hfill \\
\end{gathered} \]

 $tg\alpha  = \frac{7}{{24}}$   $0 < \alpha  < \frac{\pi }{2}$

\[\begin{gathered}
tg\alpha = \frac{{\sin \alpha }}{{\cos \alpha }} \hfill \\
\frac{{\sin \alpha }}{{\cos \alpha }} = \left. {\frac{7}{{24}}} \right| \cdot \cos \alpha \hfill \\
\sin \alpha = \frac{7}{{24}} \cdot \cos \alpha \hfill \\
\end{gathered} \]

\[\begin{gathered}
tg\alpha = \frac{{\sin \alpha }}{{\cos \alpha }} \hfill \\
\frac{{\sin \alpha }}{{\cos \alpha }} = \left. {\frac{7}{{24}}} \right| \cdot \cos \alpha \hfill \\
\sin \alpha = \frac{7}{{24}} \cdot \cos \alpha \hfill \\
{\sin ^2}\alpha + {\cos ^2}\alpha = 1 \hfill \\
{\left( {\frac{7}{{24}} \cdot \cos \alpha } \right)^2} + {\cos ^2}\alpha = 1 \hfill \\
\frac{{49}}{{576}} \cdot {\cos ^2}\alpha + {\cos ^2}\alpha = 1 \hfill \\
{\cos ^2}\alpha \left( {\frac{{49}}{{576}} + 1} \right) = 1 \hfill \\
\cos \alpha = \frac{{24}}{{25}} \hfill \\
\sin \alpha = \frac{7}{{24}} \cdot \frac{{24}}{{25}} \hfill \\
\sin \alpha = \frac{7}{{25}} \hfill \\
ctg\alpha = \frac{{\cos \alpha }}{{\sin \alpha }} = \frac{{24}}{7} \hfill \\
\end{gathered} \]