Building blocks

## Solution

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

1. the length of $OP$ is fixed but $Q$ can move along the $x$-axis as required?

2. $Q$ remains fixed but the length of $OP$ can change?

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

#### Now try this

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.

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

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

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

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

The parametric equations of a curve tell us about its shape by showing how $x$ and $y$ vary as the parameter varies. The equations may also tell us about the symmetry of the curve.

Just as there can be several Cartesian equations for the same curve, parametric equations are not unique. It may be more convenient to work with some than others, so it is worth exploring different parametrizations of the same curve.