Question

Trigonometria: me ajudem!

Answer 1

Primeiramente:

$x=r\cdot(\text{sen}~\theta)\cdot(\text{cos}~\theta)$

$x^2=r^2\cdot(\text{sen}^2~\theta)\cdot(\text{cos}^2~\theta)$

$y=r\cdot(\text{sen}~\theta)\cdot(\text{sen}~\theta)$

$y^2=r^2\cdot(\text{sen}^2~\theta)\cdot(\text{sen}^2~\theta)$

Assim:

$x^2+y^2=r^2\cdot(\text{sen}^2~\theta)\cdot(\text{cos}^2~\theta)+r^2\cdot(\text{sen}^2~\theta)\cdot(\text{sen}^2~\theta)$

Colocando $r^2\cdot(\text{sen}^2~\theta)$ em evidência, temos:

$x^2+y^2=r^2\cdot(\text{sen}^2~\theta)[\text{cos}^2~\theta+\text{sen}^2~\theta]$

Pela relação fundamental, $\text{sen}^2~\theta+\text{cos}^2~\theta=1$

Substituindo:

$x^2+y^2=r^2\cdot(\text{sen}^2~\theta)\cdot1=r^2\cdot(\text{sen}^2~\theta)$

Note que:

$z=r\cdot(\text{cos}~\theta)$

$z^2=r^2\cdot(\text{cos}^2~\theta)$

Assim:

$x^2+y^2+z^2=r^2\cdot(\text{sen}^2~\theta)+r^2\cdot(\text{cos}^2~\theta)$

Colocando $r^2$ em evidência:

$x^2+y^2+z^2=r^2\cdot[(\text{sen}^2~\theta)+(\text{cos}^2~\theta)]=r^2\cdot1=r^2$

Letra A

Answer 2

$x=rsen\theta cos \theta\\ y=rsen \theta sen \theta\\ z=rcos \theta\\ \\ x^2=r^2sen^2 \theta cos^2 \theta\\ y^2=r^2sen^2 \theta sen^2 \theta\\ z^2=r^2 cos^2 \theta\\ \\ x^2+y^2+z^2=r^2(sen^2 \theta cos^2 \theta+sen^2 \theta sen^2 \theta+ cos^2 \theta)=\\ \\ r^2[sen^2 \theta(cos^2 \theta+sen^2 \theta)+cos^2 \theta]=\\ \\ r^2[sen^2 \theta+cos^2 \theta]=r^2$