Stuck on space curves for vector valued functions

Question

I'm working through the James Stewart Calculus text to prep for school. I'm stuck at this particular point.

How would you sketch the graph for the parametric equations: $x = \cos t$, $y = \sin t$, and $z = \sin 5t$? I understand that if it were the case that $z=t$, I'd merely get a helix around the $z$-axis, as $x$ and $y$ form an ellipse. However, I cannot make the leap to solve more exotic problems such as the problem posed or even the case when $z = \ln(t)$.

Some help and a push in the right direction would be appreciated.

Answer 1

Here is an animation I made that might help. The left is a plot of $(\cos(t),\sin(t),\sin(5t))$ and the right is a plot of $\sin(5t)$.

For the case of $(\cos(t),\sin(t),\ln(t))$, here is the corresponding animation:

As a sanity check, note that in each animation, you can see that the point on the circle makes its first full revolution as $t=2\pi\approx 6.28$.

Mathematica code for my (and anyone else's) future reference:

size = 1.5

slices = 150

Slice[t_,z_] := {Show[ParametricPlot3D[{Cos[2 Pi*s], Sin[2 Pi*s], z}, 
{s, 0, 1}, PlotRange -> {{-size, size}, {-size, size}, {-size, size}}],
Graphics3D[{PointSize[Large], Point[{Cos[t], Sin[t], z}]}]], 
Show[Plot[Sin[5 s], {s, 0, 2 Pi}, Ticks -> {{0, 2 Pi/5, 4 Pi/5, 6 Pi/5, 
8 Pi/5, 2 Pi}}], Graphics[{PointSize[Large], Point[{t, Sin[5 t]}]}]]}

NewSlice[t_,z_] := {Show[ParametricPlot3D[{Cos[2 Pi*s], Sin[2 Pi*s], z},
{s, 0, 1}, PlotRange -> {{-size, size}, {-size, size}, {-2, 2}}], 
Graphics3D[{PointSize[Large], Point[{Cos[t], Sin[t], z}]}]], 
Show[Plot[Log[s], {s, 0.5, 8}, PlotRange -> {{0, 8}, {-1, 2}}, 
AspectRatio -> 1/2], Graphics[{PointSize[Large], Point[{t, Log[t]}]}]]}

Export["sin.gif", Table[Slice[2 Pi*t/slices, Sin[5*2 Pi*t/slices]], {t, 
0,slices}], "DisplayDurations" -> 0.15]

Export["ln.gif",Table[NewSlice[t, Log[t]], {t, 0.5, 7.5, 7/slices}], 
"DisplayDurations" -> 0.15]

Answer 2

Hint: Note that the $x$ and $y$ coordinates trace out a circle. As they do, the $z$ coordinate goes through $5$ sinusoidal cycles.