I have a map which I want to show is surjective. (It is only one small part of a much larger problem involving constructing a homeomorphism from the space $[([0,1]\times S^{n-1})/(\{1\}\times S^{n-1})]/(\{0\}\times S^{n-1})$.
If that notation is unfamiliar that's OK because it is irrelevant to my problem below:
I want to show that the map $g:[0,1]\times S^{n-1}\to S^n$ defined by $g(t,x) = \cos(\pi t)e_{n+1} + \sin(\pi t)x$ is surjective.
Here, $S^{n-1} = \{x\in \mathbb{R}^{n} : |x| = 1\}$ and $S^{n}$ is defined in the only consistent way. Also, by $e_{n}$ I mean the $n$th standard basis vector for Euclidean space.
The beginning of my approach is this:
Let $y\in S^{n}$. Then write $y = \Sigma_{i=1}^{n+1}y_{i}e_{i}$. By linear independence, I know immediately that if I want $g(t,x) = y$, I need to have $t = \cos^{-1}(y_{n+1})/\pi$ (chosen from $[0,\pi]$).
After typing this out I'm sort of having an idea, but since I've already typed the entire question out now I might as well post it. I'll post back with any updates. Thanks for any hints or suggestions you can give!
Update: I got it. Take $x = (y - y_{n+1}e_{n+1})$ and scale appropriately.
Sorry to trouble the crowd with this. I guess I just needed to organize the thinking process to figure it out!