Why is it so that a unit quaternion $t$ can be written as $t=\cos(\theta)+u\sin(\theta)$?

Question

Why is it so that a unit quaternion $t$ can be written as $t=\cos(\theta)+u\sin(\theta)$?

This question stems from Stillwell's Naive Lie Theory where he states that a quaternion $t$ of absolute value $1$, has a real part and an imaginary part (in $\mathbb{R}i+\mathbb{R}j+\mathbb{R}k$). This means that we can write $t$ as

$t=\cos(\theta)+u\sin(\theta)$

where $u$ is a unit vector in $\mathbb{R}i+\mathbb{R}j+\mathbb{R}k$

I wanted to understand this more clearly and any more justification on this would be appreciated.

I know we have

$t=a+bi+cj+dk =a+\frac{1}{r}r(bi+cj+dk)$

where $r^{2}=b^{2}+c^{2}+d^{2}$. If we set $q=(\frac{1}{r}(bi+cj+dk))$, we then have that $q^{2}=-1$. So we can conclude that $t=a+r\sqrt{-1}$ where $q=\sqrt{-1}$ We then can obtain that $e^{t}=e^{a}(\cos({r})+ q\sin({r}))$.

Answer 1

Every vector is a scalar times a unit vector: $\bf v$ equals $\|\bf v\|$ times $\bf v/\|v\|$.

Since $a^2+(b^2+c^2+d^2)=1$ we may write $a=\cos\theta$ and $\sqrt{b^2+c^2+d^2}=\sin\theta$ for some $\theta$. Then

$$b{\bf i}+c{\bf j}+d{\bf k}=\color{Blue}{\sqrt{b^2+c^2+d^2}}\color{Purple}{\frac{b{\bf i}+c{\bf j}+d{\bf k}}{\sqrt{b^2+c^2+d^2}}}=(\color{Blue}{\sin\theta})\color{Purple}{\bf u}$$

where $\bf u$ is the unit vector obtained by normalizing $b{\bf i}+c{\bf j}+d{\bf k}$. And so

$$a+b{\bf i}+c{\bf j}+d{\bf k}=\cos\theta+(\sin\theta){\bf u}. $$