Question

Se Ф é um angulo agudo tal que senФ = 1/3, calcule sen(2Ф) e cos(2Ф):

Answer 1

$\sin(2\Phi)=2\sin(\Phi)\cos(\Phi)\\\\\cos(2\Phi)=\cos^2(\Phi)-\sin^2(\Phi)\\\\\\\sin^2(\Phi)+\cos^2(\Phi)=1\\\\\dfrac19+\cos^2(\Phi)=1\\\\\cos(\Phi)=\sqrt{1-\dfrac19}\\\\\cos(\Phi)=\dfrac{\sqrt8}{3}\\\\\\\sin(2\Phi)=2\sin(\Phi)\cos(\Phi)=2\left(\dfrac13\right)\left(\dfrac{\sqrt8}{3}\right)=\dfrac{2\sqrt8}{9}\\\\\\\cos(2\Phi)=\cos^2(\Phi)-\sin^2(\Phi)=\dfrac89-\dfrac19=\dfrac79$