Question

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

Answer 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:

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

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

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

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

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

$M = \frac{ 1 }{ 6}$