In $\triangle ABC$,with usual notations,if $b=3$,$c=8$,and $m\angle A=60^{\circ}$,then the circumradius of the triangle is . . . . . . units.

In a $\Delta ABC$,if $b = 2$,$C = 60^\circ$,and $c = \sqrt{6}$,then $a =$

With the usual notation,in $\Delta ABC$,if $\angle A + \angle B = 120^{\circ}$ and $a : b = (\sqrt{3} + 1) : (\sqrt{3} - 1)$,then the ratio $\angle A : \angle B$ is

In a triangle $ABC$,let $AB = \sqrt{23}$,$BC = 3$,and $CA = 4$. Then the value of $\frac{\cot A + \cot C}{\cot B}$ is:

If the angles of a triangle are in the ratio $4:1:1$,then the ratio of the longest side to its perimeter is

In $\triangle ABC$, with usual notations, if $a = 4$, $b = 5$, and $c = 6$, then $\cos A$ and $\cos C$ are calculated using the Law of Cosines. Find the value of $\cos C$ and $\cos A$ to determine the relationship between $\angle C$ and $\angle A$.