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• Notes

What curve is represented by the following parametric equation?

$x=\cos (t), y=\sin (t), 0 \leq t \leq 2 \pi$

$\cos ^{2} x+\sin ^{2} x=1$

(1) $x^{2}+y^{2}=(\cos (t))^{2}+(\sin (t))^{2}=\cos ^{2}(t)+\sin ^{2}(t)=1$

(2) $t=0 \Rightarrow x=\cos (0)=1 \quad y=\sin (0)=0 \quad(x, y)=(1,0)$

$\quad(x, y)=(\cos t, \sin t)$

What curve is reprsented by the following parametric equations?

$X=\sin (2 t), y=\cos (2 t), 0 \leq t \leq 2 \pi$

(1) $x^{2}+y^{2}=(\sin (2 t))^{2}+(\cos (2 t))^{2}=\sin ^{2}(2 t)+\cos ^{2}(2 t)=1$

$(x, y)$

(2) $t=0 \Rightarrow x=\sin (2(0))=\sin (0)=0$

$y=\cos (2(0))=\cos (0)=1$

$(x, y)=(0,1)$

what are the curve represnted by the following parametric equations?

$\begin{array}{l}{\mathrm{X}=\mathrm{t}} \\ {\mathrm{Y}=t^{2}}\end{array}$

$\Rightarrow y=x^{2}$

$\begin{array}{l}{X=e^{t}} \\ {Y=e^{2 t}}\end{array}$

$x^{2}=\left(e^{t}\right)^{2}=e^{t} \cdot e^{t}=e^{2 t}$

$\Rightarrow y=x^{2}$

$\begin{array}{l}{X=t^{2}} \\ {y=t^{4}}\end{array}$

$\Rightarrow y=x^{2}$

$\begin{array}{l}{\mathrm{X}=\sin (\mathrm{t})} \\ {\mathrm{Y}=\sin ^{2}(t)}\end{array}$

$\Rightarrow y=x^{2}$