Mejia Daniel
06/06/2023 · Primary School
Si sen \( \beta=-\frac{\sqrt{63}}{8} \) y \( \beta \in\left(\frac{3 \pi}{2} ; 2 \pi\right) \), determina el valor de \( \cos \left(\frac{\beta}{2}\right) \)
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Para determinar el valor de \( \cos \left(\frac{\beta}{2}\right) \) dado que \( \sin \beta = -\frac{\sqrt{63}}{8} \) y \( \beta \in \left(\frac{3\pi}{2}; 2\pi\right) \), podemos usar la identidad de medio ángulo para el coseno:
\[
\cos \left(\frac{\beta}{2}\right) = \pm \sqrt{\frac{1 + \cos \beta}{2}}
\]
Primero, necesitamos encontrar \( \cos \beta \). Sabemos que:
\[
\sin^2 \beta + \cos^2 \beta = 1
\]
Sustituyendo \( \sin \beta \):
\[
\left(-\frac{\sqrt{63}}{8}\right)^2 + \cos^2 \beta = 1
\]
Calculamos \( \sin^2 \beta \):
\[
\frac{63}{64} + \cos^2 \beta = 1
\]
Despejamos \( \cos^2 \beta \):
\[
\cos^2 \beta = 1 - \frac{63}{64} = \frac{1}{64}
\]
Por lo tanto, \( \cos \beta \) puede ser:
\[
\cos \beta = \pm \frac{1}{8}
\]
Dado que \( \beta \) está en el cuarto cuadrante (donde \( \cos \) es positivo), tenemos:
\[
\cos \beta = \frac{1}{8}
\]
Ahora sustituimos \( \cos \beta \) en la fórmula del coseno del ángulo medio:
\[
\cos \left(\frac{\beta}{2}\right) = \sqrt{\frac{1 + \frac{1}{8}}{2}} = \sqrt{\frac{\frac{8}{8} + \frac{1}{8}}{2}} = \sqrt{\frac{\frac{9}{8}}{2}} = \sqrt{\frac{9}{16}} = \frac{3}{4}
\]
Dado que \( \frac{\beta}{2} \) estará en el intervalo \( \left(\frac{3\pi}{4}; \pi\right) \), donde el coseno es negativo, tomamos el valor negativo:
\[
\cos \left(\frac{\beta}{2}\right) = -\frac{3}{4}
\]
Por lo tanto, el valor de \( \cos \left(\frac{\beta}{2}\right) \) es:
\[
\boxed{-\frac{3}{4}}
\]
Quick Answer
El valor de \( \cos \left(\frac{\beta}{2}\right) \) es \( -\frac{3}{4} \).
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