Parametric equations Parametric equations express a set of related quantities as explicit functions of an independent parameter. This gives greater flexibility when describing curves, surfaces and motion. For a curve in the plane, x=f(t) and y=g(t), where t is the parameter and often represents time. Applications - **Motion and kinematics:** the position of a particle moving in a plane can be described by x(t) and y(t). - **Graphics and animation:** parametric equations are used to draw curves and animate objects. - **Engineering and physics:** parametric representations describe complex shapes and motion in fields such as aerodynamics and structural analysis. - **Robotics:** the path of a robot arm or the trajectory of a drone can be described parametrically. Converting parametric equations to Cartesian form Eliminate the parameter t to obtain one equation involving only x and y. The method depends on the form of the parametric equations. **Non-trigonometric form** | **Trigonometric form** Solve one or both equations for t in terms of x or y. | Identify the trigonometric ratios in both equations, such as sin t, cos t and tan t. Substitute that expression into the other equation. | Choose an identity relating the ratios, such as sin2t+cos2t=1, tan2t+1=sec2t, or 1+cot2t=cosec2t. Simplify to obtain the relationship between x and y. | Substitute into the identity and simplify the relationship between x and y.

Parametric Equations and Cartesian Form

Junior College 1

Parametric equations

Parametric equations express a set of related quantities as explicit functions of an independent parameter. This gives greater flexibility when describing curves, surfaces and motion.

For a curve in the plane, \(x=\mathrm{f}(t)\) and \(y=\mathrm{g}(t)\), where \(t\) is the parameter and often represents time.

Applications

Motion and kinematics: the position of a particle moving in a plane can be described by \(x(t)\) and \(y(t)\).
Graphics and animation: parametric equations are used to draw curves and animate objects.
Engineering and physics: parametric representations describe complex shapes and motion in fields such as aerodynamics and structural analysis.
Robotics: the path of a robot arm or the trajectory of a drone can be described parametrically.

Converting parametric equations to Cartesian form

Eliminate the parameter \(t\) to obtain one equation involving only \(x\) and \(y\). The method depends on the form of the parametric equations.

Non-trigonometric form	Trigonometric form
Solve one or both equations for \(t\) in terms of \(x\) or \(y\).	Identify the trigonometric ratios in both equations, such as \(\sin t\), \(\cos t\) and \(\tan t\).
Substitute that expression into the other equation.	Choose an identity relating the ratios, such as \(\sin^2t+\cos^2t=1\), \(\tan^2t+1=\sec^2t\), or \(1+\cot^2t=\mathrm{cosec}^2t\).
Simplify to obtain the relationship between \(x\) and \(y\).	Substitute into the identity and simplify the relationship between \(x\) and \(y\).

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