# 4.10 Parametric equations

If you’ve ever been to a science museum, then you may have seen a kind of device where if you turn a handle connected to a cog, that cog spins a bunch of other cogs. Although all the cogs spin at different rates, they’re all ultimately driven by the cog which you’re spinning.

This is a bit like how parametric equations work - we create a "parameter" (the cog which you spin, and often named $t$) and then $x$ and $y$ (or whatever the axes are called) are a bunch of other cogs connected to the initial cog.

For example, we can write the equation of the unit circle in terms of $x$ and $y$.

$x^{2}+y^{2}=1$ (4.115)

But we could also write it as two separate equations - one for $x$ in terms of a new variable we’ll introduce, $t$, and one for $y$ in terms of $t$.

 $\displaystyle x=\cos(t)$ (4.116) $\displaystyle y=\sin(t)$ (4.117)

We can get from the parametric equations (the ones in terms of $t$) to the Cartesian equations using a little algebra. Adding together $x^{2}$ and $y^{2}$ gives this equation.

$x^{2}+y^{2}=\cos^{2}(t)+\sin^{2}(t)$ (4.118)

As $\cos^{2}(t)+\sin^{2}(t)=1$, the overall result is that

$x^{2}+y^{2}=1$ (4.119)

which is the Cartesian equation of a circle with modulus one!