Take a look at triangle \(OPQ\) in this diagram. Explain why \(P\) has the coordinates given.

Imagine we move \(P\) by increasing \(\angle QOP\), whilst keeping \(PQ\) perpendicular to the \(x\)-axis.

How could \(P\) move if

the length of \(OP\) is fixed but \(Q\) can move along the \(x\)-axis as required?

\(Q\) remains fixed but the length of \(OP\) can change?

In each case, can you express the coordinates of \(P\) in terms of \(\angle QOP\) ?

If we express the coordinates of \(P\) in terms of \(\angle QOP\) then we obtain parametric equations for the path that \(P\) traces out.

We can manipulate the equations to eliminate the parameter, giving a Cartesian equation for the curve, but it can sometimes be helpful to work with parametric equations — it may make it easier to sketch the curve or to find its gradient.

Sketch the curves or lines defined by the following parametric equations. Mark on your sketch any key values of \(t\) and the direction in which the curve is traced out as \(t\) increases.

\(x=2t\) and \(y=3t-1\) for \(t\in \mathbb{R}.\)

\(x=t^3\) and \(y=t^2\) for \(-2\leq t\leq 2.\)

\(x=3\sin t\) and \(y=3\cos t\) for \(0< t < \pi.\)

For each curve, find a Cartesian equation (an equation that only involves \(x\) and \(y\)).