Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. ( Round your answers to one decimal place.)

*a* = 28,    *c* = 44,    ∠*A* = 129°

To solve for all possible triangles that satisfy the given conditions using the Law of Sines, we will follow these steps:

### Given:
- \( a = 28 \)
- \( c = 44 \)
- \( \angle A = 129^\circ \)

We need to find the other angles \( \angle C \) and \( \angle B \), and the side \( b \) of the triangle.

### Step 1: Apply the Law of Sines
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

We first use this to solve for \( \sin C \).

\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]

Substitute the known values:

\[
\frac{28}{\sin(129^\circ)} = \frac{44}{\sin C}
\]

Calculate \( \sin(129^\circ) \):

\[
\sin(129^\circ) \approx 0.7431
\]

Now, solve for \( \sin C \):

\[
\frac{28}{0.7431} = \frac{44}{\sin C}
\]

\[
37.68 = \frac{44}{\sin C}
\]

\[
\sin C = \frac{44}{37.68}
\]

\[
\sin C \approx 1.168
\]

### Step 2: Analyze for Possible Triangles
Since \( \sin C = 1.168 \) is greater than 1, this is impossible in a triangle. Therefore, no solution exists for this case.

### Conclusion:
There is **no triangle** that satisfies the given conditions because the calculated value of \( \sin C \) is greater than 1, which is impossible.

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Let me know if you'd like to explore more details or have any questions!

### Related Questions:
1. What are the possible values for \( \sin C \) in a triangle?
2. How does the Law of Sines help determine the number of possible triangles?
3. Why is it important to check the sine values when using the Law of Sines?
4. How would the problem change if \( \angle A \) were smaller than 90°?
5. Can a triangle exist if one of its angles is larger than 180°?

### Tip:
Always check the value of the sine function when solving trigonometric equations, as it must lie between -1 and 1.

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (Round your answers to one decimal place.) a = 28, c = 44, ∠A = 129°

Learn how to use the Law of Sines to determine if a triangle can exist with sides a = 28, c = 44, and angle A = 129°. This trigonometry problem involves checking sine values for triangle feasibility. Suitable for students in Grades 10-12.

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