Eliminating the Parameter from Trigonometric Equations
Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem.
Applying the general equations for conic sections, we can identify [latex]\frac{{x}^{2}}{16}+\frac{{y}^{2}}{9}=1[/latex] as an ellipse centered at [latex]\left(0,0\right)[/latex]. Notice that when [latex]t=0[/latex] the coordinates are [latex]\left(4,0\right)[/latex], and when [latex]t=\frac{\pi }{2}[/latex] the coordinates are [latex]\left(0,3\right)[/latex]. This shows the orientation of the curve with increasing values of [latex]t[/latex].
Eliminate the parameter from the given pair of parametric equations and write as a Cartesian equation:
[latex]x\left(t\right)=2\cos t[/latex] and [latex]y\left(t\right)=3\sin t[/latex].
When we are given a set of parametric equations and need to find an equivalent Cartesian equation, we are essentially “eliminating the parameter.” However, there are various methods we can use to rewrite a set of parametric equations as a Cartesian equation. The simplest method is to set one equation equal to the parameter, such as [latex]x\left(t\right)=t[/latex]. In this case, [latex]y\left(t\right)[/latex] can be any expression. For example, consider the following pair of equations.
Rewriting this set of parametric equations is a matter of substituting [latex]x[/latex] for [latex]t[/latex]. Thus, the Cartesian equation is [latex]y={x}^{2}-3[/latex].
Use two different methods to find the Cartesian equation equivalent to the given set of parametric equations.
Method 1. First, let’s solve the [latex]x[/latex] equation for [latex]t[/latex]. Then we can substitute the result into the [latex]y[/latex] equation.
