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  1. Given an explicit function $y = \sin(x) + \cos(x)$, how to convert it to the respective parametric functions $x = f_1(t)$, $y = f_2(t)$?

  2. Given parametric functions $x = \sin(t)$ and $y = \cos(t)$, how to obtain the respective implicit function $f(x,y) = 0$?

  3. Given parametric functions $x = 1+2t$ and $y = 3+4t$, how to obtain the respective implicit function $f(x,y) = 0$?

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    There are not unique answers to your questions, though in your case there are perhaps _standard_ answers. Are you looking for parametrizations having any particular properties? – 2017-01-24

1 Answers 1

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  1. You can choose for example $x$ as parameter, which leads to :

$$\cases{x=t\cr y=\sin(t)+\cos(t)}$$

  1. It is well known that $\sin^2(t)+\cos^2(t)=1$ for all $t\in\mathbb{R}$. So this curve (I pretend to ignore which curve it is ...) is included in the one with implicit equation :

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

and it should be verified that te reverse inclusion is true (provided that $t$ can take any real value, or at least any value in some $[a,a+2\pi)$).

  1. You have to "eliminate" $t$ between those two equations. The first one gives you : $t=\frac{x-1}{2}$. Putting that in the second one leads to :

$$y=3+2(x-1)$$