Finding parametric equations for the tangent line at a point on a curve

Question

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$

I understand that in order to find the solution, I need to use partial derivatives. However, the method in my textbook works for simpler problems -- I seem to be making a calculation error when I try to apply the method to this problem.

Can anyone suggest how to approach this problem?

I found a very similar problem and solution here, but the solution by the person who answered is hard for me to follow. Unfortunately, I get stuck at the line where he subtracts $\frac{\pi}{6}$ from $\pi$ within the trigonometric functions.

Here is the "simple" method that I was originally using.

Any sincere help would be appreciated. Thank you.

Answer 1

So $\textbf{r}(t) = \left< \cos t, \sin t, t \right>$. Then $\textbf{r}'(t) = \left<-\sin t, \cos t, 1 \right>$. So $t = -4 \pi/6$. So $\textbf{r'}(-\frac{4 \pi}{6}) = \left<-\sin( -\frac{4 \pi}{6}), \cos \left( -\frac{4 \pi}{6} \right), 1 \right>$. So the equation of the tangent line would be

$$x = \cos(-\frac{4 \pi}{6})+t\left(-\sin\left( -\frac{4 \pi}{6}\right)\right)$$ $$y = \sin(-\frac{4 \pi}{6})+t\left(\cos \left(-\frac{4 \pi}{6} \right) \right)$$ and $$z = -\frac{4 \pi}{6}+t$$

Answer 2

I'm pretty sure that the answer to which you linked on Yahoo! Answers is just wrong. The parametric equations $x=\cos t$, $y=\sin t$, $z=t$ describe a spiral on a cylinder of radius 1 coaxial with the $z$-axis. Since there is only 1 parameter, these parametric equations cannot describe a 2-dimensional surface. The method used in your second link seems appropriate—the direction vector of the tangent line at any point on $\langle x(t),y(t),z(t)\rangle=\langle\cos t,\sin t,t\rangle$ is $\langle x'(t),y'(t),z'(t)\rangle=\cdots$ (no partial derivatives needed) and you know a point on the line, so you can write a parametric equation for the tangent line.