4.10 Parametric equations ‣ Chapter 4 Concrete algebra ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd‣ Chapter 4 Concrete algebra ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd

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!