Suppose there are two vectors $\vec{x}, \vec{y} \in \mathbb{R}^n$. The angle $\theta$ between the two vectors can be represented as $$ \theta = \arccos{\frac{\vec{x} \cdot \vec{y}}{||\vec{x}||\cdot||\vec{y}||}} $$
Now, I am curious if there is a way to represent $\theta$ using $\arcsin$ and the dot product between the two vectors. In particular, the reason I began thinking about this is because $$ \arcsin{\frac{\vec{x} \cdot \vec{y}}{||\vec{x}||\cdot||\vec{y}||}} $$ will always fall within $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
$$\cos\theta = {\frac{\vec{x} \cdot \vec{y}}{||\vec{x}||\cdot||\vec{y}||}}$$
but $\cos(\theta)=\sin\left(\frac{\pi}{2} -\theta\right)$ and then
$$\sin\left(\frac{\pi}{2} -\theta\right) = {\frac{\vec{x} \cdot \vec{y}}{||\vec{x}||\cdot||\vec{y}||}} \Rightarrow \frac{\pi}{2} -\theta=\arcsin\left(\frac{\vec{x} \cdot \vec{y}}{||\vec{x}||\cdot||\vec{y}||}\right)$$
$$\theta=\frac{\pi}{2} -\arcsin\left(\frac{\vec{x} \cdot \vec{y}}{||\vec{x}||\cdot||\vec{y}||}\right)$$