Question

4. Se sen a = 2/3 e π/2 ≤ a ≤ π, determine os valores de:
a. sen 2a
b. cos 2a
c. tg 2a

Answer 1

$\boxed{\begin{array}{l}\sf sen(a)=\dfrac{2}{3}\implies sen^2(a)=\dfrac{4}{9}\\\sf cos^2(a)=\dfrac{9}{9}-\dfrac{4}{9}=\dfrac{5}{9}\\\sf cos(a)=-\sqrt{\dfrac{5}{9}}=-\dfrac{\sqrt{5}}{3}\\\tt a) ~\sf sen(2a)=2sen(a)cos(a)\\\sf sen(2a)=2\cdot\bigg(\dfrac{2}{3}\bigg)\cdot\bigg(-\dfrac{\sqrt{5}}{3}\bigg)\\\sf sen(2a)=-\dfrac{4\sqrt{5}}{9}\\\tt b)~\sf cos(2a)=cos^2(a)-sen^2(a)\\\sf cos(2a)=\dfrac{5}{9}-\dfrac{4}{9}\\\sf cos(2a)=\dfrac{1}{9}\\\tt c)~\sf tg(2a)=\dfrac{sen(2a)}{cos(2a)}\\\sf tg(2a)=\dfrac{-\frac{4\sqrt{5}}{\backslash\!\!\!9}}{\frac{1}{\backslash\!\!\!9}}\\\sf tg(2a)=-4\sqrt{5}\end{array}}$