Determining equation of the tangent to the surface with tricky powers

Question

I have a surface parametrised by $$ x = u\sin^3v,\quad y=u\cos^3v,\quad z=u$$ and I'm trying to find the tangent plane at $(x_0, y_0, z_0) = (\sqrt2, -\sqrt2, 4)$ using this formula: $$(x-x_0, y-y_0, z-z_0)\cdot \mathbf n(u_0,v_0)=0$$

I found the normal $ \mathbf n $ but am having difficulty finding $v_0$.

My attempt:

$ x_0=u_0\sin^3v_0 $

$ \sin^3v_0 = \frac{\sqrt2}{4} $

Is this the correct method?

Answer 1

You know that: \begin{align*} \frac{\sqrt{2}}{4} = \sin^3(v)\\ \frac{-\sqrt{2}}{4} = \cos^3(v) \end{align*} Hence we know that $\tan(v)^3 = -1 \Leftrightarrow \tan(v)=-1$. The two solutions to this equation is $-\frac{\pi}{4}$ and $\pi - \frac{\pi}{4}$, where the latter is the solution that you are looking for. The rest you already explained how to do.