Multivariable Calc issues showing curve lies on cylinder

[marquitos]

Consider the space curve x = cos t, y = sin t, z = sin^2 t.
(1) Without plotting this curve, show that this curve lies on the cylinder x^2 + y^2 = 1.
(2) Plot this curve (without the cylinder), then use the appropriate rotations to see the
planar projections on the xy-plane, the xz-plane and the yz-plane.
(3) Now, nd the equations of each of the 3 planar projections, plot them and compare to
your work on question 2 to con rm your answers.

Honestly i don't have a clue what to do any help would be nice, i think i might have to put X^2+y^2=1 into sins and cosines with respect to theta but i could be completely wrong and even if i did that i don't know where to go so please anything would be great thank you.

 

[HallsofIvy]

Consider the space curve x = cos t, y = sin t, z = sin^2 t.
(1) Without plotting this curve, show that this curve lies on the cylinder x^2 + y^2 = 1.

Just do it! What is x^2+ y^2 in terms of t?

(2) Plot this curve (without the cylinder), then use the appropriate rotations to see the
planar projections on the xy-plane, the xz-plane and the yz-plane.

I presume you are to use some kind of graphing software?

(3) Now, nd the equations of each of the 3 planar projections, plot them and compare to
your work on question 2 to conrm your answers.

In the xy-plane, z= 0 so your equations are x= cos t, y= sin t, z= 0.

In the xz-plane, y= 0 so your equations are x= cos t, y= 0, z= sin^2 t= 1- cos^2 t.

In the yz- plane, x= 0 so your equations are x= 0, y= sn t, z= sin^2 t.

Honestly i don't have a clue what to do any help would be nice, i think i might have to put X^2+y^2=1 into sins and cosines with respect to theta but i could be completely wrong and even if i did that i don't know where to go so please anything would be great thank you.