# Parametrization of the 4-dimensional unit sphere

Given the function $$x:[0,\pi]\times[0,\pi]\times[0, 2\pi]\rightarrow \mathbb{R}^4$$ where $$x(\theta_1,\theta_2,\varphi)=(\cos\theta_1,\sin\theta_1 \cos\theta_2, sin\theta_1 \sin\theta_2 \cos\varphi, \sin\theta_1 \sin\theta_2 \sin\varphi)$$, I want to show that $$\text{Im}(x)=S^3$$ where $$S^3$$ is the 4-dimensional unit sphere.

I have shown that $$\text{Im}(x)\subseteq S^3$$. But now I have problems showing the other direction.

Let $$(w,x,y,z)$$ be a coordinate on $$S^3$$. Break this quadruple into two bits, $$w$$ and $$(x,y,z)$$. Let $$l_1$$ be the length of the first of those, and $$l_2$$ be the length of the second. Then $$\sqrt{l_1^2 + l_2^2}=1$$. Therefore, set $$\cos\theta_1$$ equal to $$l_1$$ and $$\sin\theta_1$$ equal to $$l_2$$. Then divide all of $$x$$, $$y$$ and $$z$$ in $$(x,y,z)$$ by $$\sin\theta_1$$ and repeat the process.