円周の媒介変数表示

$ \forall x,y\in\R;x^2+y^2=1\iff\exist!\theta\in[0,2\pi\lbrack;x=\cos\theta\land y=\sin\theta

証明

$ x^2+y^2=1

$ \iff 0\le|x|\le1\land |y|=\sqrt{1-x^2}

$ \iff\exist!\theta\in[0,\pi];

$ \begin{dcases}x=\cos\theta\\|y|=|\sin\theta|\end{dcases}

$ \iff\exist!\theta\in[0,\pi];

$ \begin{dcases}x=\cos\theta\\y=\sin\theta\lor y=\sin(-\theta)\end{dcases}

$ \iff\exist!\theta\in]-\pi,\pi];

$ \begin{dcases}x=\cos\theta\\y=\sin\theta\end{dcases}

$ \underline{\iff\exist!\theta\in[0,2\pi\lbrack;\begin{dcases}x=\cos\theta\\y=\sin\theta\end{dcases}\quad}_\blacksquare