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$\begingroup$

Where $t$ ranges from $0$ to $\infty$.

After a bit of manipulation, I got $x^2 + y^2 = t^2$ but I have no idea how to sketch this since $t$ isn't a constant.

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    Your curve is a spiral – 2017-01-25
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    It should be a spiral, think about taking circles who get bigger and bigger as $t$ increases. – 2017-01-25
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    Yes, I know it's a spiral but I want to know how exactly I could graph it if I didn't already know what it looked like. – 2017-01-25
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    To clarify, do you 1. Understand how to graph a polar equation $r = f(\theta)$? 2. Know how to recognize when a parametric curve is a polar graph? – 2017-01-26

2 Answers 2

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Your curve is given by

$$ \alpha(t) = (t \cos t, t \sin t) = t (\cos t, \sin t). $$

As $t$ runs from $0$ to $2\pi$ the curve $(\cos t, \sin t)$ traces a circle of radius one around the origin. Multiplying by $t$ scales the point $(\cos t, \sin t)$ appropriately so that at time $t$ the resulting point makes an angle $t$ with the positive $x$-axis but lies on the circle of radius $t$ instead of the circle on radius one. Hence, you get a spiral that starts (when $t = 0$) at the origin and spirals counter clockwise as $t$ increases from $0$ to $2\pi$ and reaches $(1,0)$ when $t = 2\pi$ and then the pattern continues.

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    Thank you! This is exactly what I was looking for. – 2017-01-26
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If you are have access to any mathematical software you can visual what's going on quite nicely. I've plotted the two situations which are described in the above answer.
For a curve given by $x(t)=t \cos(t)$ and $y(t)= t\sin(t)$ in the range $0\leq t \leq 5\pi$ (just for visual purposes) you get the following:

enter image description here

Then notice for $x(t)= \cos(t)$ and $y(t)= \sin(t)$ in the range $0\leq t \leq 2\pi$ you get the following: enter image description here

Notice for the second graph as long as the range is $\geq 2\pi$ we will always get a circle. It won't spiral as in the first case because there is no factor of $t$ to scale the graph. Anyway - I always find visual aids fantastic help in remembering what's going on.