Let $X$ be a Hilbert space with scalar product $(\cdot,\cdot)$. Then for two vectors $v,w$ of norm $1$, we can interpret $(v,w)$ as an angle, so that $(v,w)=\cos(\varphi)$ for a unique angle $\varphi\in[0,\pi)$.
My question is the following: Let $\varphi'\in[0,\pi)$ and $v'\in X$ with $\|v'\|=1$ (norm induced by the scalar product) be given. Is there a vector $w'\in X$ with $(v',w')=\cos(\varphi')$?
Thank you in advance!