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To solve this problem using Cramer's rule for the cosines of angles \(\beta\) and \(\gamma\), we must first recall how Cramer's rule is applied in the context of systems of linear equations. In a triangle, we can derive relationships between the cosines of angles and the sides using the Law of Cosines and then use Cramer's rule to solve for the unknown cosines.

### Step-by-step solution

#### 1. Set up the Law of Cosines equations for the triangle:
For a triangle with sides \(a\), \(b\), and \(c\), and angles \(\alpha\), \(\beta\), and \(\gamma\), we can write the Law of Cosines as follows:

\[
c^2 = a^2 + b^2 - 2ab\cos(\gamma)
\]
\[
a^2 = b^2 + c^2 - 2bc\cos(\alpha)
\]
\[
b^2 = a^2 + c^2 - 2ac\cos(\beta)
\]

#### 2. Express the system in matrix form:
We can rearrange the above equations into a linear system involving \(\cos(\alpha)\), \(\cos(\beta)\), and \(\cos(\gamma)\). However, in this case, we focus on \(\cos(\beta)\) and \(\cos(\gamma)\).

Let’s write the system as follows:

\[
A \vec{x} = \vec{b}
\]

Where:
\[
A = \begin{pmatrix}
-2ab & 2bc \\
2ac & -2ab
\end{pmatrix}
\]

We can use determinants and Cramer's rule to solve for \(\cos(\beta)\) and \(\cos(\gamma)\).

#### 3. Apply Cramer's rule:
Cramer's rule gives solutions for a system \(A \vec{x} = \vec{b}\) using determinants:

\[
x_i = \frac{\text{det}(A_i)}{\text{det}(A)}
\]

Where \(\text{det}(A)\) is the determinant of the coefficient matrix \(A\), and \(\text{det}(A_i)\) is the determinant of \(A\) with the \(i\)-th column replaced by the vector on the right-hand side.

Let me know if you'd like more details on expanding this approach or further clarification! Here are a few related questions:

1. How does Cramer's rule help solve linear systems in general?
2. Can Cramer's rule be applied to non-square matrices? Why or why not?
3. How do we calculate the determinant of a \(2 \times 2\) matrix?
4. What are the applications of the Law of Cosines in trigonometry?
5. Can you generalize this process to find all three cosines in any triangle?

**Tip:** When solving systems of equations involving trigonometric terms, always check if simplifications can be made before applying formal rules like Cramer's.

Use Cramer's rule to obtain similar formulas for cos(β) and cos(γ).

Learn how to use Cramer's rule to derive formulas for cos(β) and cos(γ) in a triangle using the Law of Cosines and matrix methods. Suitable for students at the undergraduate level.

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