The problem provides the lengths of two sides of a triangle and the angle opposite to one of the sides. The goal is to determine which law (Law of Sines or Law of Cosines) should be used to find the angle that the stove makes with the sink.

1. Problem Description

The problem provides the lengths of two sides of a triangle and the angle opposite to one of the sides. The goal is to determine which law (Law of Sines or Law of Cosines) should be used to find the angle that the stove makes with the sink.

2. Solution Steps

The sides are the distance between the stove and sink (1.3 m) and the distance between the sink and refrigerator (2.2 m). The angle between the refrigerator and the stove is $31.9^{\circ}$. We want to find the angle that the stove makes with the sink, which is opposite to the side with length 2.2 m.

Let $a$ be the distance between the sink and the refrigerator, so $a = 2.2$ m.

Let $b$ be the distance between the stove and the sink, so $b = 1.3$ m.

Let angle $C$ be the angle at the refrigerator, so $C = 31.9^{\circ}$.

We are looking for angle $A$, at the stove.

We have two sides and an angle. The Law of Cosines could be used to first find the length of the third side $c$. The Law of Cosines is $c^2 = a^2 + b^2 - 2ab \cos(C)$. After that, the Law of Sines could be used: $\frac{\sin A}{a} = \frac{\sin C}{c}$ or the Law of Cosines could also be used $a^2 = b^2 + c^2 - 2bc \cos A$.

Alternatively, the Law of Sines could be used directly to find angle $A$, using the formula $\frac{\sin A}{a} = \frac{\sin C}{c}$, but we don't have $c$. Also, we can use Law of Cosines directly to find angle A, we know two sides and the included angle $C$.

Law of Cosines states $c^2 = a^2 + b^2 - 2ab \cos{C}$, where $a$, $b$ and $c$ are sides of the triangle and $C$ is the angle opposite side $c$.

We know two sides and the included angle C. Therefore we can use the Law of Cosines to find the third side c. After having three sides, we can find the other angles using either the Law of Sines or the Law of Cosines.

3. Final Answer

Law of Cosines can be used to find the third side. Then Law of Sines or Law of Cosines can be used to find the remaining angles.