Question

Con la información de la figura halle tg²α​

Answer 1

Respuesta:

$tg^2(\alpha)=\frac{1}{4}$

Explicación paso a paso:

AC= k

$tg(\alpha)=\frac{3}{k}\\ tg(2\alpha)=\frac{8}{k}\\ \\ propiedad:\\tg(2 \theta)=\frac{2tg(\theta)}{1-tg^2(\theta)}$

aplicamos la propiedad

$tg(2\alpha)=\frac{2tg(\alpha)}{1-tg^2(\alpha)}\\\\ \frac{8}{k}=\frac{2(\frac{3}{k}) }{1-(\frac{3}{k})^2 }\\\\ \frac{8}{k}=\frac{\frac{6}{k} }{1-\frac{9}{k^2} }\\\\ \frac{8}{k}=\frac{\frac{6}{k} }{\frac{k^2-9}{k^2} }\\\\ \frac{8}{k}=\frac{6(k^2)}{k(k^2-9)} \\\\ 8=\frac{6(k^2)}{k^2-9}\\\\ 8(k^2-9)=6k^2\\ 8k^2-72=6k^2\\ 8k^2-6k^2=72\\ 2k^2=72\\\ k^2=\frac{72}{2}\\\ k^2=36\\ k=\sqrt{36}\\ k=6$

hallando $tg^2(\alpha)$

$(\ tg(\alpha)\ )^2=(\frac{3}{6})^2\\\\ tg^2(\alpha)=\frac{3^2}{6^2}\\\\ tg^2(\alpha)=\frac{9}{36}\\\\ tg^2(\alpha)=\frac{1}{4}$