the path of a particle in the xy-plane is vector r = (cos2t, sint) for t for all [-pi/2, pi] where t represents time. Sketch the path. Is it a smooth curve?

Question

How do i sketch the path? do i just plug in random points between [-pi/2, pi]? how would i connect it together? and how do i know if it's a smooth curve? Thank you.

Answer 1

If that pair of coordinates represents (X,Y), then the easiest way is to do pretty much what you already suggested, but do it systematically in small increments. That is, work out a set of points as follows. -pi/2 is approximately -1.57, so

t=-1.57, X=cos(2t)=-1, Y=sin(t)=-1
t=-1.4, X=cos(2t)=-0.942, Y=sin(t)=-0.985
t=-1.3, X=cos(2t)=-0.856, Y=sin(t)=-0.963
etc etc. Then just plot X vs Y. If you've got access to a spreadsheet you'll find it very quick to do that.

Just one question though: are you sure that interval isn't [-pi/2, +pi/2]? If so, then yes, it's a smooth curve between those limits. But if you're correct about the upper limit for t being pi, then the curve is rather more complicated: it runs smoothly from (-1,1) through (0,0), up to (-1,1) and then bounces back on itself again to (0,0), overwriting itself in the process. If I had to bet, I'd guess that the question has been copied down wrong.