Answer
$ \angle C=63^{\circ}$
$b=1116.9$
and
$c=999$
Work Step by Step
I. Need to find the last angle.
$22^{\circ}+95^{\circ}+\angle C=180^{\circ}$
$\implies \angle C=63^{\circ}$
II. Use law of sines formula to find $a$ and $b$
$\dfrac{\sin 22^{\circ}}{420}=\dfrac{\sin 95^{\circ} }{b}=\dfrac{\sin 63^{\circ} }{c}$
This implies that
$b=\dfrac{420 \sin 95^{\circ} }{\sin 22^{\circ}}=1116.9$
and
$c=\dfrac{420 \sin 63^{\circ} }{\sin 22^{\circ}}=999$
