Applying the trigonometric ratios to a general triangle.
$${h = b \cdot \sin{\alpha}}$$
and
$${h = a \cdot \sin{\beta}}$$
Equating the two equations:
$${a \cdot \sin{\beta} = b \cdot \sin{\alpha}}$$
Rearranging the equations
$${\frac{a}{\sin{\alpha}} = \frac{b}{ \sin{\beta}}}$$
Similarly, you can show that
$${\frac{a}{\sin{\alpha}} = \frac{c}{\sin{\gamma}}}$$
and
$${\frac{b}{\sin{\beta}} = \frac{c}{\sin{\gamma}}}$$
the sine rule
$${\frac{a}{\sin{\alpha}} = \frac{b}{\sin{\beta}} = \frac{c}{\sin{\gamma}}}$$
or
$${\frac{a}{b} = \frac{\sin{\alpha}}{\sin{\beta}}}$$
applications
Aufgabe 60 and 61 on Aufgabenfuchs
CIMT - exercises (same as on cosine rule page)