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The Law of Sines is a powerful tool in trigonometry that relates the lengths of sides of a triangle to the sines of its angles. This rule is particularly useful for solving non-right triangles, where traditional trigonometric ratios of sine, cosine, and tangent fail to apply directly. According to the Law of Sines, for any triangle with angles A, B, and C opposite to sides a, b, and c, respectively, the following proportion holds:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

This means that the ratio of a side length to the sine of its opposite angle is the same for all three sides of the triangle.

To solve for a missing side using the Law of Sines, you reorganize the proportion to isolate the unknown side.

• For example, if you are solving for side \( b \), and you know side \( a \) and angles \( A \) and \( B \), you can use: \( b = \frac{a \cdot \sin B}{\sin A} \)

In the exercise provided, this formula was applied after calculating angle A, which helps to find the other sides accurately.

In any triangle, the sum of its internal angles is always \( 180^{\circ} \). This fundamental property helps to solve triangles when you know two angles and need to find the third. Remember:

• \( A + B + C = 180^{\circ} \)

To find the unknown angle when two angles are given, subtract the sum of those two angles from \( 180^{\circ} \). For instance, in our exercise, we knew angles B and C, calculated as:

• \( A = 180^{\circ} - 58.4^{\circ} - 22.2^{\circ} = 99.4^{\circ} \)

This simple calculation offers a direct way to move forward in solving the entire triangle when combined with the Law of Sines or Cosines.

Solving a triangle involves systematic steps that start from identifying given and unknown elements. Here's a typical approach used in the exercise:

• Identify Known and Unknowns: List what you are given and what you need to find. In our task, side \( a \) and angles \( B \) and \( C \) were known.


• Calculate the Third Angle: Use the property that the sum of the angles is \( 180^{\circ} \). This helps in finding one unknown angle, which is crucial in applying trigonometric laws.


• Apply the Law of Sines: Once angles are known, use the Law of Sines to find the missing sides. This provides a straightforward method to calculate the lengths based on the angles and one known side.

Following these steps ensures you can methodically work through solving any triangle problem, leading to a comprehensive understanding of each element involved. Always check your work by ensuring all internal angles sum to \( 180^{\circ} \) and verifying calculated sides logically fit into the triangle.