Тригонометријске функције полуугла 1

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Пр.1  Израчунати тригонометријске функције угла $\alpha=\frac{\pi }{{12}} $

Пр.2 Проверити

 \[\frac{{1 - \sin \alpha  + \cos \alpha }}{{1 - \sin \alpha  - \cos \alpha }} =  - ctg\frac{\alpha }{2}\]

Пр.1 

\[\begin{gathered}
\sin \frac{\pi }{{12}} = \sin \frac{{\frac{\pi }{6}}}{2} = \sqrt {\frac{{1 - \cos \frac{\pi }{6}}}{2}} = \sqrt {\frac{{1 - \frac{{\sqrt 3 }}{2}}}{2}} = \hfill \\
= \sqrt {\frac{{\frac{{2 - \sqrt 3 }}{2}}}{{\frac{2}{1}}}} = \sqrt {\frac{{2 - \sqrt 3 }}{4}} = \frac{{\sqrt {2 - \sqrt 3 } }}{2} \hfill \\
\cos \frac{\pi }{{12}} = \cos \frac{{\frac{\pi }{6}}}{2} = \sqrt {\frac{{1 + \cos \frac{\pi }{6}}}{2}} = \sqrt {\frac{{1 + \frac{{\sqrt 3 }}{2}}}{2}} = \hfill \\
= \sqrt {\frac{{\frac{{2 + \sqrt 3 }}{2}}}{{\frac{2}{1}}}} = \sqrt {\frac{{2 + \sqrt 3 }}{4}} = \frac{{\sqrt {2 + \sqrt 3 } }}{2} \hfill \\
tg\frac{\pi }{{12}} = \frac{{\sin \frac{\pi }{{12}}}}{{\cos \frac{\pi }{{12}}}} = \frac{{\frac{{\sqrt {2 - \sqrt 3 } }}{2}}}{{\frac{{\sqrt {2 + \sqrt 3 } }}{2}}} = \frac{{\sqrt {2 - \sqrt 3 } }}{{\sqrt {2 + \sqrt 3 } }} \cdot \frac{{\sqrt {2 + \sqrt 3 } }}{{\sqrt {2 + \sqrt 3 } }} = \hfill \\
= \frac{{\sqrt {\left( {2 - \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)} }}{{2 + \sqrt 3 }} = \frac{{\sqrt {4 - 3} }}{{2 + \sqrt 3 }} = \frac{1}{{2 + \sqrt 3 }} \cdot \frac{{2 - \sqrt 3 }}{{2 - \sqrt 3 }} = \hfill \\
= \frac{{2 - \sqrt 3 }}{{4 - 3}} = 2 - \sqrt 3 \hfill \\
ctg\frac{\pi }{{12}} = \frac{{\cos \frac{\pi }{{12}}}}{{\sin \frac{\pi }{{12}}}} = \frac{{\frac{{\sqrt {2 + \sqrt 3 } }}{2}}}{{\frac{{\sqrt {2 - \sqrt 3 } }}{2}}} = 2 + \sqrt 3 \hfill \\
\end{gathered} \]

Пр.2

\[\begin{gathered}
\frac{{1 - \sin \alpha + \cos \alpha }}{{1 - \sin \alpha - \cos \alpha }} = - ctg\frac{\alpha }{2} \hfill \\
\frac{{2{{\cos }^2}\frac{\alpha }{2} - \sin \alpha }}{{2si{n^2}\frac{\alpha }{2} - \sin \alpha }} = - ctg\frac{\alpha }{2} \hfill \\
\frac{{2{{\cos }^2}\frac{\alpha }{2} - 2\sin \frac{\alpha }{2} \cdot \cos \frac{\alpha }{2}}}{{2si{n^2}\frac{\alpha }{2} - 2\sin \frac{\alpha }{2} \cdot \cos \frac{\alpha }{2}}} = - ctg\frac{\alpha }{2} \hfill \\
\frac{{2\cos \frac{\alpha }{2}\left( {\cos \frac{\alpha }{2} - \sin \frac{\alpha }{2}} \right)}}{{ - 2sin\frac{\alpha }{2}\left( {\cos \frac{\alpha }{2} - \sin \frac{\alpha }{2}} \right)}} = - ctg\frac{\alpha }{2} \hfill \\
\frac{{\cos \frac{\alpha }{2}}}{{ - sin\frac{\alpha }{2}}} = - ctg\frac{\alpha }{2} \hfill \\
\end{gathered} \]