Finding a Rectangular Equation for a Plane Curve Represented Parametrically

Find a rectangular equation of the curve whose parametric equations are \[ \begin{equation*} x( t) =R\, \cos t \qquad y( t) =R\, \sin t \end{equation*} \]

where \(R>0\) is a constant. Graph the plane curve and indicate its orientation.

Solution The presence of the sine and cosine functions in the parametric equations suggests using the Pythagorean Identity \(\cos ^{2}t+\sin ^{2}t=1\). Then \[ \begin{array}{rcl@{\quad}crcl} \left( \dfrac{x}{R}\right) ^{2}+\left( \dfrac{y}{R}\right) ^{2}& =&1 &\qquad\color{#0066A7}{\cos \;t = \tfrac{x}{R}\quad \sin t=\dfrac{y}{R}}\\ x^{2}+y^{2}& =& R^{2} \end{array} \]

The graph of the rectangular equation is a circle with center at the origin and radius \(R\). In the parametric equations, as the parameter \(t\) increases, the points \((x,y)\) on the circle are traced out in the counterclockwise direction, as shown in Figure 3.