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To find side \( a = BC \) in this triangle, we can use the Law of Cosines. The given information provides us:

- Angle \( \angle A = 16.25^\circ \)
- Side \( AB = 36.52 \, \text{cm} \)
- Side \( AC = 29.43 \, \text{cm} \)

The Law of Cosines states:
\[
a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
\]
Where:
- \( a = BC \) (side we want to find)
- \( b = AB = 36.52 \, \text{cm} \)
- \( c = AC = 29.43 \, \text{cm} \)
- \( \angle A = 16.25^\circ \)

Now, applying the Law of Cosines:

\[
a^2 = 36.52^2 + 29.43^2 - 2 \times 36.52 \times 29.43 \times \cos(16.25^\circ)
\]

Let me calculate this for you:

\[
a^2 = 1333.1904 + 866.0049 - 2 \times 36.52 \times 29.43 \times \cos(16.25^\circ)
\]
\[
a^2 = 2199.1953 - 2145.6476 \times \cos(16.25^\circ)
\]
\[
a^2 = 2199.1953 - 2055.628
\]
\[
a^2 = 143.5673
\]

Taking the square root:

\[
a = \sqrt{143.5673} \approx 11.98 \, \text{cm}
\]

Therefore, side \( BC \) is approximately \( 11.98 \, \text{cm} \).

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen understanding:
1. How would the Law of Sines help in this problem?
2. What if the given angle was an obtuse angle? How would that change the solution?
3. Can you use the Pythagorean theorem here? Why or why not?
4. What is the significance of using the Law of Cosines for non-right triangles?
5. How can you find the other two angles of this triangle?

**Tip:** Always double-check that you are applying the correct formula, especially when distinguishing between the Law of Sines and Law of Cosines.

Learn how to find the side length of a triangle using the Law of Cosines. Given angle A = 16.25°, side AB = 36.52 cm, and side AC = 29.43 cm, calculate the length of side BC. This problem involves trigonometry concepts suitable for Grades 10-12 students.

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