mfb said:Do you have to simplify the equation? You found an equation y=f(x).
trollcast said:I actually just realized that I've misread the equations so I can see how to solve it now.
The equations should be:
$$x=tan(t)-1\qquad y=cot^2(t)+1$$
So the solution would be:
$$tan(t)=x+1$$
$$y=\frac{1}{tan^2(t)}$$
$$y=\frac{1}{(x+1)^2}$$
Dick said:What happened to the '+1' in ##y=cot^2(t)+1##?
trollcast said:Oops, so it should be:
$$x=tan(t)-1\qquad y=cot^2(t)+1\\
tan(t)=x+1\\
y - 1=\frac{1}{tan^2(t)}\\
y=\frac{1}{(x+1)^2} +1\\
y=\frac{(x+1)^2+1}{(x+1)^2}$$
A cartesian equation is a mathematical equation that expresses the relationship between two variables using x and y as the independent and dependent variables, respectively.
A parametric equation is a set of equations that describe the motion of a point in terms of one or more parameters. It is often used to represent curves and surfaces.
To convert a parametric equation to a cartesian equation, you can use the substitution method. Replace the parameters with expressions in terms of x and y, and then eliminate the parameters by solving for x and y.
Parametric equations are useful because they can describe complex curves and surfaces that are difficult to represent with cartesian equations. They also allow for more flexibility in defining the shape and motion of an object.
Not all parametric equations have a corresponding cartesian equation. Some curves and surfaces are only defined parametrically and cannot be represented by a single cartesian equation.