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To find the angle \( A \) using the cosine rule, we proceed as follows:

### Cosine Rule:
\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\]
Here:
- \( a = 12 \) (side opposite angle \( A \)),
- \( b = 8 \),
- \( c = 10 \).

### Step 1: Substitute the values into the formula.
\[
\cos A = \frac{8^2 + 10^2 - 12^2}{2 \cdot 8 \cdot 10}
\]

### Step 2: Calculate the numerator.
\[
8^2 = 64, \quad 10^2 = 100, \quad 12^2 = 144
\]
\[
\cos A = \frac{64 + 100 - 144}{160}
\]
\[
\cos A = \frac{20}{160}
\]

### Step 3: Simplify.
\[
\cos A = 0.125
\]

### Step 4: Find angle \( A \) by taking the inverse cosine.
\[
A = \cos^{-1}(0.125)
\]
Using a calculator:
\[
A \approx 82.819^\circ
\]

### Final Answer:
\[
A = 82.819^\circ
\]

Let me know if you'd like further explanation or clarification! Here are some related questions to explore:

1. How does the cosine rule apply to obtuse angles in a triangle?
2. What are some practical uses of the cosine rule outside of mathematics?
3. How can we verify our result using the sine rule?
4. What if the triangle sides are not labeled? How should we set up the cosine rule?
5. How can this rule help in solving real-world navigation or engineering problems?

**Tip:** Always double-check your calculations when applying trigonometric formulas to ensure accuracy!

Use the cosine rule to find the value of A. Give your answer correct to three decimal places.

Learn how to find the angle A in a triangle using the cosine rule. This problem involves substituting values for sides 8, 10, and 12, solving for cos A, and applying inverse trigonometric functions. Suitable for Grades 10-12 students.

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