Question

seja x um arco do primeiro quadrante, se secx 3/2, então qual o valor de senx, tgx e cossecx

Answer 1

$\Large\boxed{\begin{array}{l}\sf sec(x)=\dfrac{3}{2}\implies cos(x)=\dfrac{2}{3}\\\sf tg^2(x)+1=sec^2(x)\\\sf tg^2(x)+1=\dfrac{9}{4}\\\\\sf tg^2(x)=\dfrac{9}{4}-1\\\\\sf tg^2(x)=\dfrac{5}{4}\\\\\sf tg(x)=\dfrac{\sqrt{5}}{\sqrt{4}}\\\\\sf tg(x)=\dfrac{\sqrt{5}}{2}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf sen(x)=cos(x)\cdot tg(x)\\\\\sf sen(x)=\dfrac{\diagup\!\!\!2}{3}\cdot\dfrac{\sqrt{5}}{\diagup\!\!\!2}=\dfrac{\sqrt{5}}{3}\\\\\sf cossec(x)=\dfrac{3}{\sqrt{5}}\cdot\dfrac{\sqrt{5}}{\sqrt{5}}\\\\\sf cossec(x)=\dfrac{3\sqrt{5}}{5}\end{array}}$