How to convert the expression to parametric?

Question

I have the expression

$$ y=\sqrt{R^2-x^2} $$

Know, I know that the result of the parametric conversion will be:

$$ x=R\cos(t), y=R\sin(t)$$

And yet I don't really understand how to get from one to the other.

Hope someone can clarify if for me ?

Answer 1

Given: $y=\sqrt{R^2-x^2}\iff y^2=R^2-x^2\iff x^2+y^2=R^2$

Substitute $x=R\cos(t)\ \ \forall \ \ t\in[0, 2\pi]$ to get $y$ coordinate as follows $$y=\sqrt{R^2-x^2}=\sqrt{R^2-(R\cos(t))^2}=\sqrt{R^2(1-\cos^2(t))}=R\sin(t)$$

Similarly, substitute $y=R\sin(t)\ \ \forall \ t\in\left[0, 2\pi\right]$ to get $x$ coordinate as follows $$x=\sqrt{R^2-y^2}=\sqrt{R^2-(R\sin(t))^2}=\sqrt{R^2(1-\sin^2(t))}=R\cos(t)$$

Answer 2

The switch between

$$ y^2 + x^2 = R^2 \ (1)$$

and $$ x= R\cos(t) \ y=R\sin(t)\ (2)$$

is as follow.

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$$ 1 \to 2$$It is a choice of function, note that we can invert the $x$ and $y$ parametric function it also works.

$$ 2 \to 1$$

Just take $x^2$ and $y^2$ of $(2)$ and use $\cos^2+\sin^2=1$.