
Use the Law of Sines to find the indicated side x in the following figure. (Round your answer to one decimal place.) A triangle is given.
The first side of length 310 is opposite the angle 98.4°.
The second side is opposite the angle 24.6°.
The third side of length x is opposite an unlabeled angle.


To solve for the unknown side \( x \) in the triangle using the Law of Sines, we apply the formula:

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

Where:
- \( a \), \( b \), and \( c \) are the lengths of the sides.
- \( A \), \( B \), and \( C \) are the angles opposite to the sides.

### Given:
- Side \( a = 310 \) opposite angle \( A = 98.4^\circ \).
- Side \( b = x \) opposite an unlabeled angle \( C \).
- Angle \( B = 24.6^\circ \) opposite side \( c \).

We can first find angle \( C \), since the sum of angles in a triangle is \( 180^\circ \):

\[
C = 180^\circ - A - B = 180^\circ - 98.4^\circ - 24.6^\circ = 57^\circ
\]

Now we can use the Law of Sines to solve for \( x \) (side \( b \)):

\[
\frac{310}{\sin(98.4^\circ)} = \frac{x}{\sin(57^\circ)}
\]

Solving for \( x \):

\[
x = \frac{310 \times \sin(57^\circ)}{\sin(98.4^\circ)}
\]

Now, let's compute the value of \( x \).The length of side \( x \) is approximately \( 262.8 \) units.

Would you like any further explanation or details on this solution?

Here are 5 related questions for practice:
1. How would you apply the Law of Cosines to solve for a side in a triangle?
2. What is the Law of Sines, and how does it work for non-right triangles?
3. How do you find an unknown angle in a triangle using the Law of Sines?
4. Can the Law of Sines be used in right triangles as well?
5. What are the steps to solve for an unknown side in a triangle when given all three angles?

**Tip**: Always check if the triangle is solvable by ensuring that the sum of angles is exactly \(180^\circ\).

Use the Law of Sines to find the indicated side x in the following figure. (Round your answer to one decimal place.) A triangle is given. The first side of length 310 is opposite the angle 98.4°. The second side is opposite the angle 24.6°. The third side of length x is opposite an unlabeled angle.

Learn how to use the Law of Sines to find the unknown side x in a triangle. In this problem, one side is 310 units opposite a 98.4° angle, and another side is opposite a 24.6° angle. Follow a step-by-step approach to solving for x and understanding key trigonometric concepts. Suitable for students in Grades 10-12.

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