# A calculus of the absurd

#### 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 $$y$$.

$$x^2 + y^2 = 1$$

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 $$t$$.

\begin{align} & x = \cos (t) \\ & y = \sin (t) \end{align}

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.

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

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

$$x^2 + y^2 = 1$$

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