I recently learned in math class how to convert the rectangular equation of a circle into a parametric equation:
$x^2+y^2=1$
$\cos^2(t)+\sin^2(t)=1$
$x^2+y^2=\cos^2(t)+\sin^2(t)$
$x^2=\cos^2(t)$
$y^2=\sin^2(t)$
$x=\cos(t)$
$y=\sin(t)$
I don't understand the logic behind going from $x^2+y^2=\cos^2(t)+\sin^2(t)$ to $x^2=\cos^2(t)$ and $y^2=\sin^2(t)$. My teacher explained through an analogy:
$4+3=7$
$(1+3)+(1+2)=7$
Therefore, $1+3=4$ and $1+2=3$.
My teacher also said that I would know to pair $x^2$ with $\cos^2(t)$ because cosine is related to x in the unit circle and that $y^2$ pairs with $\sin^2(t)$ for the same reason.
My problem with this explanation is that I can also write these equations:
$4+3=7$
$(1+1)+(2+3)=7$
In this case, $4\neq1+1$ and $3\neq2+3$, so the analogy doesn't seem to apply to all cases.
Additionally, the graph of the parametric function is the same (except for direction) when $x$ is paired with $\sin$ or $\cos$, so I don't understand why that matters. The reasoning also doesn't make sense to me because I don't see how the unit circle definition of $\sin$ and $cos$ can decide how they pair with two variables.
Is the logic behind this step just knowing that $x^2=\cos^2(t)$ and $y^2=\sin^2(t)$ can be simplified to $x^2+y^2=\cos^2(t)+\sin^2(t)$ from the steps to convert a parametric circle equation to rectangular? Or is there a way to do this step without having to already know that it's allowed?
Best Answer
Let
$ x = sin(t + a), $
$y = cos(t + a).$
Clearly $x^2 + y^2 = 1$.