Answer
$AB=10.2$.
$BC=6.3$.
Work Step by Step
The given values are
$AC=8$ and $\angle B=52^{\circ}$.
The triangle is shown in the image file.
Use trigonometric ratios.
$\Rightarrow \sin B =\frac{AC}{AB}$
Multiply both sides by $\frac{AB}{\sin B}$.
$\Rightarrow \frac{AB}{\sin B}\cdot \sin B =\frac{AB}{\sin B}\cdot\frac{AC}{AB}$
Simplify.
$\Rightarrow AB=\frac{AC}{\sin B}$
Substitute all values.
$\Rightarrow AB=\frac{8}{\sin 52^{\circ}}$
Simplify.
$AB=10.2$.
Now use
$\Rightarrow \tan B =\frac{AC}{BC}$
Multiply both sides by $\frac{BC}{\tan B}$.
$\Rightarrow \frac{BC}{\tan B} \cdot \tan B =\frac{BC}{\tan B} \cdot \frac{AC}{BC}$
Simplify.
$\Rightarrow BC =\frac{AC}{\tan B}$
Substitute all values.
$\Rightarrow BC =\frac{8}{\tan 52^{\circ}}$
Simplify.
$BC=6.3$.
