In $\triangle ABC$, if $a = 125, \; b = 123, \; c = 62$, find the sines of half the angles and the sines of the angles.
In any $\triangle ABC$,
$\sin \left(\dfrac{A}{2}\right) = \sqrt{\dfrac{\left(s - b\right) \left(s - c\right)}{b c}}$, $\;$ $\sin \left(\dfrac{B}{2}\right) = \sqrt{\dfrac{\left(s - c\right) \left(s - a\right)}{c a}}$, $\;$ $\sin \left(\dfrac{C}{2}\right) = \sqrt{\dfrac{\left(s - a\right) \left(s - b\right)}{a b}}$
$\sin A = \dfrac{2}{bc} \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$, $\;$ $\sin B = \dfrac{2}{ca} \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$, $\;$ $\sin C = \dfrac{2}{ab} \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$
$s = \dfrac{a + b + c}{2} = $ semi perimeter of $\triangle ABC$
Given: $\;$ $a = 125, \; b = 123, \; c = 62$
$s = \dfrac{125 + 123 + 62}{2} = 155$
$\begin{aligned}
\sin \left(\dfrac{A}{2}\right) & = \sqrt{\dfrac{\left(155 - 123\right) \left(155 - 62\right)}{123 \times 62}} \\\\
& = \sqrt{\dfrac{32 \times 93}{123 \times 62}} = \sqrt{\dfrac{16}{41}} = \dfrac{4}{\sqrt{41}}
\end{aligned}$
$\begin{aligned}
\sin \left(\dfrac{B}{2}\right) & = \sqrt{\dfrac{\left(155 - 62\right) \left(155 - 125\right)}{62 \times 125}} \\\\
& = \sqrt{\dfrac{93 \times 30}{62 \times 125}} = \sqrt{\dfrac{9}{25}} = \dfrac{3}{5}
\end{aligned}$
$\begin{aligned}
\sin \left(\dfrac{C}{2}\right) & = \sqrt{\dfrac{\left(155 - 125\right) \left(155 - 123\right)}{125 \times 123}} \\\\
& = \sqrt{\dfrac{30 \times 32}{125 \times 123}} = \sqrt{\dfrac{64}{25 \times 41}} = \dfrac{8}{5 \sqrt{41}}
\end{aligned}$
$\begin{aligned}
\sin A & = \dfrac{2}{123 \times 62} \sqrt{155 \left(155 - 125\right) \left(155 - 123\right) \left(155 - 62\right)} \\\\
& = \dfrac{1}{123 \times 31} \sqrt{155 \times 30 \times 32 \times 93} \\\\
& = \dfrac{1}{123 \times 31} \times 4 \times 5 \times 6 \times 31 = \dfrac{40}{41}
\end{aligned}$
$\begin{aligned}
\sin B & = \dfrac{2}{62 \times 125} \sqrt{155 \left(155 - 125\right) \left(155 - 123\right) \left(155 - 62\right)} \\\\
& = \dfrac{1}{31 \times 125} \sqrt{155 \times 30 \times 32 \times 93} \\\\
& = \dfrac{1}{31 \times 125} \times 4 \times 5 \times 6 \times 31 = \dfrac{24}{25}
\end{aligned}$
$\begin{aligned}
\sin C & = \dfrac{2}{125 \times 123} \sqrt{155 \left(155 - 125\right) \left(155 - 123\right) \left(155 - 62\right)} \\\\
& = \dfrac{2}{125 \times 123} \sqrt{155 \times 30 \times 32 \times 93} \\\\
& = \dfrac{2}{125 \times 123} \times 4 \times 5 \times 6 \times 31 = \dfrac{496}{1025}
\end{aligned}$