Question

sendo α a medida de um ângulo agudo tal que tg α= 2, calcule sen α e cos α

Answer 1

sena/cosa = 2/1

sen²a/cos²a = 4/1

sen²a+cos²a/cos²a = 4+1/1

1/cos²a= 5

cos²a =1/5

cosa = √1/5 = √5/5 ✓

sena=√1-cos²a=√1-(1/5)=√4/5 = 2/√5 =

2√5 /5 ✓

Answer 2

$\large\boxed{\begin{array}{l}\sf tg(\alpha)=2\implies tg^2(\alpha)=4\\\sf 1+tg^2(\alpha)=sec^2(\alpha)\\\sf 1+4= sec^2(\alpha)\\\sf sec^2(\alpha)=5\\\sf sec(\alpha)=\sqrt{5}\\\sf \dfrac{1}{cos(\alpha)}=\sqrt{5}\\\\\sf cos(\alpha)=\dfrac{1}{\sqrt{5}}=\dfrac{\sqrt{5}}{5}\\\\\sf sen(\alpha)=cos(\alpha)\cdot tg(\alpha)\\\sf sen(\alpha)=\dfrac{\sqrt{5}}{5}\cdot2\\\\\sf sen(\alpha)=\dfrac{2\sqrt{5}}{5}\end{array}}$