si 16=8^tgalfa, siendo alfa un ángulo agudo, calcular el valor de: M=(Cot(45°+alfa/2)) tg(alfa/2)​

Respuestas

Respuesta dada por: jabche
1

Hola (。•̀ᴗ-)✧

Respuesta:

M = \frac{ 1  }{ 6}

Explicación paso a paso:

16=8^{  \tg(\alpha )}

 {2}^{4} =2^{ 3 \tg(\alpha )}

4=3 \tg(\alpha )

 \tg(\alpha ) =  \frac{4}{3}

\star \: \: \tg (2 \cdot \frac{\alpha }{2} ) = \frac{2 \tg (\frac{\alpha }{2} ) }{ 1 - {\tg }^2 ( \frac{\alpha }{2})}</p><p>

\: \: \tg (\alpha ) = \frac{2 \tg (\frac{\alpha }{2} ) }{ 1 - {\tg }^2 ( \frac{\alpha }{2})}</p><p>

\: \:  \frac{4}{3}  = \frac{2 \tg (\frac{\alpha }{2} ) }{ 1 - {\tg }^2 ( \frac{\alpha }{2})}</p><p>

\: \:  4 - 4{\tg }^2 ( \frac{\alpha }{2}) = 6\tg (\frac{\alpha }{2} )

4{\tg }^2 ( \frac{\alpha }{2})  + 6\tg (\frac{\alpha }{2} ) - 4 = 0

La  \: raiz  \: positiva:  \:  \tg  ( \frac{ \alpha }{2} ) =  \frac{1}{2}

Calcular:

M = \cot ( 45^{\circ}+ \frac{\alpha }{2} ) \cdot \tg (\frac{ \alpha }{2} )

M = \frac{ 1 -\tg ( {45}^{ \circ})  \cdot  \tg ( \frac{ \alpha }{2})  }{\tg ( {45}^{ \circ}) +  \tg ( \frac{ \alpha }{2}) }  \cdot \tg (\frac{ \alpha }{2} )

M = \frac{ 1 -1 \cdot   \frac{1}{2}  }{1+  \frac{1}{2} }  \cdot  \frac{1}{2}

M = \frac{ \frac{1}{2}  }{ \frac{3}{2} }  \cdot  \frac{1}{2}

M = \frac{ 1  }{ 3}  \cdot  \frac{1}{2}

M = \frac{ 1  }{ 6}

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