A calculus of the absurd
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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 \(y\).
\(\seteqnumber{0}{4.}{114}\)
\begin{equation}
x^2 + y^2 = 1
\end{equation}
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 \(t\).
\(\seteqnumber{0}{4.}{115}\)
\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.
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\begin{equation}
x^2 + y^2 = \cos ^2 (t) + \sin ^2 (t)
\end{equation}
As \(\cos ^2 (t) + \sin ^2 (t) = 1\), the overall result is that
\(\seteqnumber{0}{4.}{118}\)
\begin{equation}
x^2 + y^2 = 1
\end{equation}
which is the Cartesian equation of a circle with modulus one!