Question

on which triangle can the law of cosines be used to find the length of an unknown side? law of cosines: $a^{2}=b^{2}+c^{2}-2bc\cos(a)$

Explanation:

Step1: Recall law - of - cosines condition

The law of cosines $a^{2}=b^{2}+c^{2}-2bc\cos(A)$ is used when we know two - side lengths and the included angle between them in a triangle.

Step2: Analyze first triangle

In the first triangle with angles $36^{\circ},57^{\circ},87^{\circ}$ and side length $QS = 12$, we don't have two - side lengths and the included angle given to use the law of cosines.

Step3: Analyze second triangle

In the second triangle, we know side lengths $QS = 12$ and $RS=7$, and the included angle $\angle S=57^{\circ}$. We can use the law of cosines to find the length of side $s$ (opposite to $\angle S$). For example, if we want to find $s$, we can use $s^{2}=12^{2}+7^{2}-2\times12\times7\times\cos(57^{\circ})$.

Answer:

The second triangle (the one with side lengths 12 and 7 and included angle $57^{\circ}$)