Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve.
$ r(t) = \langle t , \sin t , 2 \cos t \rangle $
see work for graph
we're gonna look at the projections onto each of the coordinate planes and then we're gonna magically use that to figure out what this is a picture of. Okay so first we know X's. T. And why is the sign of T. So that's the sign of X. So in the Z. X. Sorry in the xy plane. What? I'm just gonna go okay I'm pretty sure that's not zero and I'm just gonna go from from 0 to 2 pi, okay here's what its projection is. So it's like it's shadow there's a light over at Z. Equals three and it's pointing right at it and that's the shadow that you see in the xy plane. Yeah. Okay then we know X. S. T. And Z. Is to co sign T. So that's to co sign X. Okay so in the Z. Explain it looks like the co sign. So now you're standing at Y. equals three and you flash a light over there and you see that. Okay now what about in the Xz plane? Well yeah. Mhm. Oh I just did that. I mean in the Y. Z. Playing. Well somehow I have to. Mhm. Somehow I have to figure out how Synnex and to cosign XR related. Oh yeah or sign tion to co sign T. Let's do that. Okay well I know that the sine squared of T. Plus the uh if I square Z. I get four coastline square T. So let's put it put it before here for sine square T. Plus four coastline square T. Equals four. I just know that right because I know sine squared plus coastline squared is one. So if I multiply everything by ford then I get it equal to four. We'll sign square T. That is why squared So four y squared for coastline square T. That Z squared equals four. Okay so that's why squared over one plus Z squared over four equals one. So in the Z. Y. Plane up to on the Z. Down to on the Z. one on the Y. It looks like this. Okay so now let's put it all together. So when you look at it from the Z. Y. Plane it looks like an ellipse. So it's rolling down or up and around this like this moving in the X. Direction like this. Why direction. I don't even know how to say it. Okay so in the Z. Y. Plane we have a cylinder. Okay. Uh that's oblong because of the two. See you at the scene by playing. Yeah. And the Z. Y. Plane. We have any lips, sorry? So since we don't have anything about X in that part of it we have this cylinder that we're going to be traveling along. Okay? And it's going to travel along in that in the way in this way. Okay I don't know how to I don't know how to do that. Besides plot some points. So let's just plot some points. So remember we had um T. Cy. Inti to co sign T. So if we plug in zero we get 002. So that's where starting 002 on the Z. So we're starting here. Just let me get another colour going here. Okay now the next most convenient thing to put in would be pi over two unless you want but unless you want a lot of points let's just try pi over two. However 2 10. So five or 2 on the X. one on the why? Yeah. All right so we're right here so we've come from here to there. Well I'm pretending like that's pi over two right there on the X. Axis and then Pie I get pie. Sign up. I zero goes on by minus one so minus two. So now when we get out here to X equals pi why is back to zero again? But now Z. Is down here at negative two. So now we're down here so see it going around us here and then it's going to come back out so the X. Is getting bigger and the Z. And the Y. Have to stay on this. So it's something like that. So it's spiraling spiraling around and moving that way because of this X equal steve. Okay but it's going around this elliptical thing because of these two. Okay hope that