We are given a triangle formed by a stove, a sink, and a refrigerator. The distance between the stove and the sink is 1.3 meters, the distance between the sink and the refrigerator is 2.2 meters, and the angle formed at the refrigerator is 31.9 degrees. We need to determine which law (Law of Sines or Law of Cosines) we should use to find the angle that the stove makes with the sink.

1. Problem Description

We are given a triangle formed by a stove, a sink, and a refrigerator. The distance between the stove and the sink is 1.3 meters, the distance between the sink and the refrigerator is 2.2 meters, and the angle formed at the refrigerator is 31.9 degrees. We need to determine which law (Law of Sines or Law of Cosines) we should use to find the angle that the stove makes with the sink.

2. Solution Steps

The Law of Sines states that for any triangle:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

where $a, b, c$ are the side lengths and $A, B, C$ are the angles opposite those sides.

The Law of Cosines states that for any triangle:

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

In this problem, we know two sides and one angle (side-angle-side). Let $a$ be the distance between the sink and the refrigerator (2.2 meters), $b$ be the unknown distance between the stove and the refrigerator, $c$ be the distance between the stove and the sink (1.3 meters), and $A$ be the angle at the stove (unknown), $B$ be the angle at the sink, and $C$ be the angle at the refrigerator (31.9 degrees).

Since we know two sides and the included angle (the angle between those sides), we can use the Law of Cosines to find the third side $b$. After that, we can use either the Law of Sines or the Law of Cosines to find the other angles. However, since the question asks which law we *should* use, and the Law of Cosines can be directly applied to find the angle at the stove ($A$), we use the Law of Cosines.

3. Final Answer

We should use the Law of Cosines because we are given two sides and the included angle, and we want to find the angle opposite one of the known sides.

Specifically, we know sides $a = 2.2$ and $c = 1.3$, and angle $C = 31.9^\circ$. We can first use the Law of Cosines to find side $b$: $c^2 = a^2 + b^2 - 2ab \cos C$

Then we can use the Law of Cosines again to find angle $A$:

$a^2 = b^2 + c^2 - 2bc \cos A$, so $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$, which we can solve for $A$.