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To solve this problem, we need to determine the correct trigonometric law to apply, given the sides \( a = 3.5 \), \( b = 8.2 \), and \( c = 5.4 \) of a triangle. 

### Step 1: Determine the Applicable Law
1. **Law of Sines** is usually used if we have a known angle opposite a known side (angle-side pair) to find another angle or side. Here, no angles are provided, so this law is not directly applicable.
   
2. **Law of Cosines** is generally used when we know:
   - Three sides (SSS) and need to find an angle.
   - Two sides and the included angle (SAS) to find the third side.
   
Since we have all three sides \( a \), \( b \), and \( c \) but no angles, the **Law of Cosines** is appropriate for this problem.

### Step 2: Applying the Law of Cosines
The Law of Cosines states that for any triangle with sides \( a \), \( b \), and \( c \):
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
We can rearrange this to solve for \( \cos(C) \):
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Similarly, we can find \( \cos(A) \) and \( \cos(B) \) for the other angles.

Would you like a detailed calculation of each angle using the Law of Cosines?

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Here are related questions to deepen understanding:
1. How can the Law of Sines be applied when at least one angle is known?
2. When is it preferable to use the Law of Cosines over the Law of Sines?
3. What are the steps to find all angles in a triangle using the Law of Cosines?
4. How can you confirm if three given side lengths form a valid triangle?
5. What other properties can be determined in a triangle if all sides and angles are known?

**Tip:** Always double-check the applicability of each law based on known values before solving.

Determine which law applies. Then solve the triangle.
a = 3.5, b = 8.2, c = 5.4
Which law applies?
A. The law of sines
B. The law of cosines
C. Cannot be solved

Learn how to determine the applicable trigonometric law for solving a triangle given sides a=3.5, b=8.2, and c=5.4. This problem demonstrates the application of the Law of Cosines in trigonometry, ideal for high school students in Grades 10-12.

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