# Parametrization of unit sphere in $\mathbb{R}^3$

I would like to show (I'm not yet sure if it's true, though), that any vector $v\in \mathbb{R}^3$ with $\|v\| = 1$ can be written as $\left(\cos(\beta)\sin(\alpha),\; \sin(\alpha)\sin(\beta), \; \cos^2\left(\frac{\alpha}{2}\right)-\sin^2\left(\frac{\alpha}{2}\right)\right)^T$.

Hereby, $0 \leq \alpha \leq \pi$, and $0 \leq \beta < 2 \pi$

Any ideas whether it's true or how to show it?

• Don't you mean $\cos^2-\sin^2$ in the last one?. Anyways, take a look at this
– Pedro
Jul 25, 2013 at 14:15
• Change the first cos(alpha) to sin(alpha), and replace sin(alpha/2) by sin^2(alpha/2), and then it works. Jul 25, 2013 at 14:15
• Yes, editing it. Thanks! Can you give me a hint on how it works? Jul 25, 2013 at 14:16

Let $v=(v_1,v_2,v_3)$ have length 1.Let $\alpha$ be the angle between the vector $v$ and the vector $(0,0,1)$. Let $\beta$ be the angle between $(v_1,v_2,0)$ and $(1,0,0)$. We find that $v_1=sin(\alpha)cos(\beta),v_2=sin(\alpha)sin(\beta),v_3=cos(\alpha)=cos^2(\frac{\alpha}{2})-sin^2(\frac{\alpha}{2})$
• Please prefix trig.functions' names in $\LaTeX$ with a backslash, so that they render in upright font as functions' symbols: \sin → $\sin$ rather than as mutiplication of three variables, s, i & n: sin → $sin$. Same for $\cos$. Jul 14, 2016 at 8:34