Question

Eliminate the parameter. x = 3 cos t, y = 3 sin t

Answer 1

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Eliminate the parameter from the parametric equations:

$\left\{ \begin{array}{l} \mathtt{x=3\,cos\,t}\\ \mathtt{y=3\,sin\,t} \end{array} \right.\qquad\qquad\mathtt{t\in\mathbb{R}}$

The key is trying to find a relation between  x  and  y, so you can get rid of that parameter  t. See what happens if you square both equations:

$\left\{ \begin{array}{l} \mathtt{x^2=(3\,cos\,t)^2}\\ \mathtt{y^2=(3\,sin\,t)^2} \end{array} \right.\\\\\\ \left\{ \begin{array}{l} \mathtt{x^2=3^2\,cos^2\,t}\\ \mathtt{y^2=3^2\,sin^2\,t} \end{array} \right.$

Now, add them both, and you have:

$\mathtt{x^2+y^2=3^2\,cos^2\,t+3^2\,sin^2\,t}\\\\ \mathtt{x^2+y^2=3^2\cdot (cos^2\,t+sin^2\,t)\qquad\quad but~~cos^2\,t+sin^2\,t=1}\\\\ \mathtt{x^2+y^2=3^2\cdot 1}$

$\begin{array}{lcl} \!\!\!\mathtt{x^2+y^2=3^2}&\quad\longleftarrow\quad&\texttt{and now we have already}\\&&\texttt{eliminated the parameter.} \end{array}$

That equation represents a circumference, whose center is at the origin (0, 0), and has a radius of  3.


I hope this helps. =)


Tags:  eliminate parameter parametric equation curve circle circumference sine cosine sin cos trigonometric trig relation identity algebra calculus