In a triangle, two sides and the angle between them are known. Let's calculate the cotangent of the unknown angles.
Let the known data be $a, b$ and $\gamma$, and let's calculate, for example, $\operatorname{cotg} \beta$. The sine theorem does not establish a relationship among these quantities, because

$$
\frac{a}{b}=\frac{\sin \alpha}{\sin \beta}
$$

here the given $\gamma$ angle does not appear, instead $\alpha$ appears; however, this can be eliminated using the relationship

$$
\alpha=180^{\circ}-\beta-\gamma
$$

According to equation (2), $\alpha$ and $(\beta+\gamma)$ complement each other to $180^{\circ}$, therefore

$$
\sin \alpha=\sin (\beta+\gamma)=\sin \beta \cos \gamma+\cos \beta \sin \gamma
$$

Substituting this into (1)

$$
\frac{a}{b}=\cos \gamma+\operatorname{cotg} \beta \sin \gamma
$$

from which $\operatorname{cotg} \beta$ can be expressed with the known data

$$
\operatorname{cotg} \beta=\frac{a}{b \sin \gamma}-\operatorname{cotg} \gamma
$$

Similarly, we can obtain that

$$
\operatorname{cotg} \alpha=\frac{b}{a \sin \gamma}-\operatorname{cotg} \gamma
$$