Respuesta :
Para encontrar el valor de \(\tan(2x)\) dado que \(\tan(x) = -\frac{4}{3}\), podemos usar la identidad trigonométrica \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\).
Dado que \(\tan(x) = -\frac{4}{3}\), sustituimos este valor en la fórmula:
\[
\tan(2x) = \frac{2 \cdot (-\frac{4}{3})}{1 - (-\frac{4}{3})^2}
\]
Calculando:
\[
\tan(2x) = \frac{-\frac{8}{3}}{1 - \frac{16}{9}}
\]
\[
\tan(2x) = \frac{-\frac{8}{3}}{\frac{9}{9} - \frac{16}{9}}
\]
\[
\tan(2x) = \frac{-\frac{8}{3}}{\frac{-7}{9}}
\]
\[
\tan(2x) = -\frac{8}{3} \cdot \frac{-9}{7}
\]
\[
\tan(2x) = \frac{72}{21}
\]
\[
\tan(2x) = \frac{24}{7}
\]
Entonces, si \(\tan(x) = -\frac{4}{3}\), entonces \(\tan(2x) = \frac{24}{7}\).
Dado que \(\tan(x) = -\frac{4}{3}\), sustituimos este valor en la fórmula:
\[
\tan(2x) = \frac{2 \cdot (-\frac{4}{3})}{1 - (-\frac{4}{3})^2}
\]
Calculando:
\[
\tan(2x) = \frac{-\frac{8}{3}}{1 - \frac{16}{9}}
\]
\[
\tan(2x) = \frac{-\frac{8}{3}}{\frac{9}{9} - \frac{16}{9}}
\]
\[
\tan(2x) = \frac{-\frac{8}{3}}{\frac{-7}{9}}
\]
\[
\tan(2x) = -\frac{8}{3} \cdot \frac{-9}{7}
\]
\[
\tan(2x) = \frac{72}{21}
\]
\[
\tan(2x) = \frac{24}{7}
\]
Entonces, si \(\tan(x) = -\frac{4}{3}\), entonces \(\tan(2x) = \frac{24}{7}\).