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To find the angle \( A \) using the cosine rule, we can use the formula:

\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]

where:
- \( a = 5 \) (the side opposite angle \( A \)),
- \( b = 4 \) (one of the other sides),
- \( c = 6 \) (the other side).

### Step-by-Step Solution:

1. Substitute the values into the formula:

\[
\cos(A) = \frac{4^2 + 6^2 - 5^2}{2 \cdot 4 \cdot 6}
\]

2. Calculate the squares:

\[
\cos(A) = \frac{16 + 36 - 25}{48}
\]

\[
\cos(A) = \frac{27}{48}
\]

3. Simplify the fraction:

\[
\cos(A) = 0.5625
\]

4. Use the inverse cosine function to find \( A \):

\[
A = \cos^{-1}(0.5625)
\]

5. Calculate \( A \) to three decimal places:

\[
A \approx 55.771^\circ
\]

Therefore, the value of \( A \) is approximately **55.771 degrees**.

Would you like more details on any step or have other questions?

### Related Questions:
1. How do you derive the cosine rule?
2. Can the cosine rule be used for non-obtuse angles as well?
3. How would you find angle \( B \) or \( C \) using the cosine rule?
4. What would change if the triangle was obtuse?
5. How can you verify the result using the sine rule?

### Tip:
Always ensure that your calculator is set to the correct mode (degrees or radians) based on the angle measurement you're working with.

Use the cosine rule to find the value of angle A in a triangle with sides 4, 5, and 6. Give your answer correct to three decimal places.

Learn how to use the cosine rule to find angle A in a triangle with sides 4, 5, and 6. This detailed step-by-step solution covers the formula cos(A) = (b^2 + c^2 - a^2) / (2bc), and provides an answer accurate to three decimal places, approximately 55.771 degrees. Ideal for high school trigonometry students.

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