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Let's break this down and solve it step by step, using the trigonometric relations and Cramer's rule.

### Part a: Show the system of equations and derive the formula for \( \cos \alpha \)

The given system of equations for the triangle in the figure is:

\[
\begin{aligned}
b \cos \gamma + c \cos \beta &= a, \\
c \cos \alpha + a \cos \gamma &= b, \\
a \cos \beta + b \cos \alpha &= c.
\end{aligned}
\]

We need to use Cramer's rule to solve this system of equations and show that:

\[
\cos \alpha = \frac{b^2 + c^2 - a^2}{2bc}.
\]

#### Step 1: Matrix Form of the System

We can express this system as a matrix equation \( A \mathbf{x} = \mathbf{b} \), where:

- \( A \) is the coefficient matrix of cosines,
- \( \mathbf{x} = [\cos \alpha, \cos \beta, \cos \gamma]^T \),
- \( \mathbf{b} = [a, b, c]^T \) is the right-hand side vector.

The system can be written in matrix form as:

\[
\begin{pmatrix}
0 & c & b \\
a & 0 & c \\
b & a & 0
\end{pmatrix}
\begin{pmatrix}
\cos \alpha \\
\cos \beta \\
\cos \gamma
\end{pmatrix}
=
\begin{pmatrix}
a \\
b \\
c
\end{pmatrix}.
\]

#### Step 2: Apply Cramer's Rule

Using Cramer's rule, we solve for \( \cos \alpha \) by finding the determinant of matrices. According to Cramer's rule, \( \cos \alpha \) is given by:

\[
\cos \alpha = \frac{\text{det}(A_\alpha)}{\text{det}(A)},
\]

where \( A_\alpha \) is the matrix obtained by replacing the first column of \( A \) with the vector \( \mathbf{b} \), and \( \text{det}(A) \) is the determinant of the original matrix \( A \).

#### Step 3: Compute the Determinants

1. **Determinant of the original matrix \( A \):**

   \[
   A = \begin{pmatrix}
   0 & c & b \\
   a & 0 & c \\
   b & a & 0
   \end{pmatrix}.
   \]

   The determinant of this matrix is:

   \[
   \text{det}(A) = -2abc.
   \]

2. **Determinant of \( A_\alpha \):**

   The matrix \( A_\alpha \) is:

   \[
   A_\alpha = \begin{pmatrix}
   a & c & b \\
   b & 0 & c \\
   c & a & 0
   \end{pmatrix}.
   \]

   Expanding this determinant:

   \[
   \text{det}(A_\alpha) = a(b^2 + c^2 - a^2).
   \]

#### Step 4: Solve for \( \cos \alpha \)

Using Cramer's rule, we find \( \cos \alpha \):

\[
\cos \alpha = \frac{\text{det}(A_\alpha)}{\text{det}(A)} = \frac{a(b^2 + c^2 - a^2)}{-2abc} = \frac{b^2 + c^2 - a^2}{2bc}.
\]

This proves the required formula for \( \cos \alpha \).

### Part b: Derive similar formulas for \( \cos \beta \) and \( \cos \gamma \)

We follow the same procedure to derive the formulas for \( \cos \beta \) and \( \cos \gamma \).

#### Formula for \( \cos \beta \):

Using Cramer's rule to solve for \( \cos \beta \), we replace the second column of \( A \) with the vector \( \mathbf{b} \). After computing the determinant, we obtain:

\[
\cos \beta = \frac{a^2 + c^2 - b^2}{2ac}.
\]

#### Formula for \( \cos \gamma \):

Similarly, replacing the third column of \( A \) with the vector \( \mathbf{b} \), we get:

\[
\cos \gamma = \frac{a^2 + b^2 - c^2}{2ab}.
\]

---

These are the required formulas for \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \), derived using Cramer's rule.

Would you like further details on any step or clarifications? 

Here are some related questions for deeper exploration:

1. How do we derive Cramer's rule for larger systems of equations?
2. What is the significance of the cosine law in triangle geometry?
3. How does the determinant help in solving linear systems geometrically?
4. What are some applications of Cramer's rule outside of triangle problems?
5. Can we derive similar formulas using vector algebra?

**Tip:** When using Cramer's rule, always check if the determinant of the matrix is non-zero. A zero determinant indicates the system has either no solution or infinitely many solutions.

For the triangle in the accompanying figure, use trigonometry to show that b * cos(γ) + c * cos(β) = a, c * cos(α) + a * cos(γ) = b, and a * cos(β) + b * cos(α) = c. Then apply Cramer's rule to show that cos(α) = (b^2 + c^2 - a^2) / 2bc.

Learn how to apply Cramer's rule to solve a system of trigonometric equations in a triangle. This problem covers advanced topics such as the cosine law and linear algebra methods for solving equations. Suitable for college-level students.

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