So I came across this intresting question while looking at some fun math problems relating to curves in space.
Q: Determine the point at which $f(t)=\langle t, t^2,t^3\rangle$ and $g(t)=\langle \cos t, \sin t , t+1 \rangle$ intersect.
So if they do intersect at point say $\langle \cos t, \sin t , t+1 \rangle$, then $\cos^2 t= \sin t$. From here we can write $\sin^2t +\sin^t-1=0$. Solving this does not yield any familiar angles. I seem to be running into a dead end. Is there a better way?
The equation $\cos^3 t=t+1$ has only one solution, $t=0$, which is not a solution to $\cos^2(t)=\sin(t)$. So the curves do not intersect. But in general, if you discover that $\sin t = (-1+\sqrt{5})/2=\alpha$, then you can get an algebraic expression for $\cos t = \sqrt{1-\alpha^2}$. Instead of plugging in for $t$, one plugs in for $\cos t$. (I'm assuming here that $t$ is real, because the problem says "curves in space.")