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We are tasked with finding the side \( x \) using the Law of Sines. The Law of Sines states:

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

From the triangle, we know:
- Angle \( \angle A = 98.4^\circ \)
- Angle \( \angle B = 24.6^\circ \)
- Side opposite \( \angle C \) is \( 406 \) units.

### Step 1: Find \( \angle C \)
Since the angles of a triangle sum to \( 180^\circ \):

\[
\angle C = 180^\circ - (98.4^\circ + 24.6^\circ) = 57^\circ
\]

### Step 2: Apply the Law of Sines
Now, we use the Law of Sines to find \( x \), the side opposite \( \angle A \):

\[
\frac{x}{\sin(98.4^\circ)} = \frac{406}{\sin(57^\circ)}
\]

### Step 3: Solve for \( x \)
Rearrange the equation to solve for \( x \):

\[
x = \frac{406 \times \sin(98.4^\circ)}{\sin(57^\circ)}
\]

Now, I'll calculate the value of \( x \).The length of side \( x \) is approximately \( 478.9 \) units when rounded to one decimal place.

Would you like any further details or clarification?

Here are 5 related questions you could explore:
1. How does the Law of Cosines differ from the Law of Sines?
2. How can you apply the Law of Sines to non-right triangles?
3. What happens if a triangle has no solution using the Law of Sines (ambiguous case)?
4. Can the Law of Sines be used in right-angled triangles?
5. How would you solve for an angle using the Law of Sines if all sides are given?

**Tip:** Always ensure that the angles you sum in a triangle add up to 180° to avoid calculation errors.

Use the Law of Sines to find the indicated side x in the following figure. (Round your answer to one decimal place.)

Learn how to solve for side x using the Law of Sines in a triangle with angles 98.4° and 24.6°. This step-by-step solution demonstrates how to apply the Law of Sines to calculate unknown sides and angles in non-right triangles.

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