= Solution
Both segments start at $(1,1,1)$, so their direction <vectors> are $\mathbf a=(-1,-1,-1)$ and $\mathbf b=(1,1,0)$. Their <dot product> is $-2$, and their <norms> are $\sqrt3$ and $\sqrt2$. Thus
$$
\boxed{\cos\theta=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}=-\frac2{\sqrt6}=-\sqrt{\frac23}.}
$$
The angle between the rays lies in $[0,\pi]$, so its <sine> is nonnegative:
$$
\boxed{\sin\theta=\sqrt{1-\cos^2\theta}=\frac1{\sqrt3}.}
$$
The negative <cosine> correctly identifies an obtuse angle. One also obtains the <sine> from $|\mathbf a\times\mathbf b|/(|\mathbf a||\mathbf b|)=\sqrt2/\sqrt6$, using the <cross product>.
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