Question

Se a tg (a+b)=9/2 e tg b= 1/4, então quanto vale a tg a?

Answer 1

Resposta:

Tem-se a seguinte identidade trigonométrica:

$\sf tg(\alpha+\beta)=\dfrac{tg\,\alpha+tg\,\beta}{1-tg\,\alpha\,tg\,\beta}$

De acordo com o enunciado, sendo igual a 9/2 com $\sf tg\,\beta=\frac{1}{4}$, segue que:

$\sf\dfrac{tg\,\alpha+\frac{1}{4}}{1-tg\,\alpha\cdot\frac{1}{4}}=\dfrac{9}{2}$

$\sf tg\,\alpha+\dfrac{1}{4}=\dfrac{9}{2}\bigg(1-\dfrac{1}{4}tg\,\alpha\bigg)$

$\sf tg\,\alpha+\dfrac{1}{4}=\dfrac{9}{2}-\dfrac{9}{8}tg\,\alpha$

$\sf tg\,\alpha+\dfrac{9}{8}tg\,\alpha=\dfrac{9}{2}-\dfrac{1}{4}$

$\sf \dfrac{8+9}{8}tg\,\alpha=\dfrac{18-1}{4}$

$\sf \dfrac{17}{8}tg\,\alpha=\dfrac{17}{4}$

$\sf tg\,\alpha= \dfrac{8}{17}\cdot\dfrac{17}{4}$

$\sf tg\,\alpha= \dfrac{8}{4}$

$\red{\boxed{\sf tg\,\alpha= 2}}$