Find all solutions to the following triangle. (Round your answers to one decimal place. If either triangle is not possible, enter NONE in each corresponding answer blank.)
A = 116.9°, a = 43.3 cm, b = 23.2 cm
First triangle (assume B ≤ 90°):

To solve the triangle, we need to determine the other angles and the remaining side using the given information: \( A = 116.9^\circ \), \( a = 43.3 \) cm, and \( b = 23.2 \) cm.

We can use the Law of Sines to find the angle \( B \):

\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]

Rearranging to solve for \( \sin B \):

\[
\sin B = \frac{b \cdot \sin A}{a}
\]

Substituting the given values:

\[
\sin B = \frac{23.2 \cdot \sin 116.9^\circ}{43.3}
\]

Let's calculate this.The angle \( B \) is approximately \( 28.5^\circ \).

Since we assume \( B \leq 90^\circ \), this solution is valid. Next, we'll find angle \( C \) using the fact that the sum of angles in a triangle is \( 180^\circ \):

\[
C = 180^\circ - A - B
\]

Let's calculate \( C \).The angle \( C \) is approximately \( 34.6^\circ \).

Now, let's find the remaining side \( c \) using the Law of Sines:

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

Rearranging to solve for \( c \):

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

Let's calculate side \( c \).The length of side \( c \) is approximately 27.5 cm.

So, the solutions for the triangle are:
- \( B = 28.5^\circ \)
- \( C = 34.6^\circ \)
- \( c = 27.5 \) cm

If you have any further questions or need additional details, feel free to ask! 

Here are some related questions to explore:
1. How would you solve for a different triangle if angle \( B \) were greater than 90°?
2. What if the given angle \( A \) was an acute angle instead of an obtuse angle?
3. How does the Law of Cosines compare with the Law of Sines for solving triangles?
4. How would you verify if a triangle is possible given three side lengths?
5. What is the significance of the ambiguous case in the Law of Sines?
6. How would the solution change if one of the sides was the hypotenuse in a right triangle?
7. What are the possible values of \( B \) if multiple triangles were possible?
8. How does the area of this triangle compare to one with different angles but the same side lengths?

**Tip:** When solving for triangles, always check if the given information might lead to an ambiguous case, especially when dealing with the Law of Sines.

Learn how to solve a triangle with given angles and sides using the Law of Sines. Find angles B and C, and side c for a triangle with A = 116.9°, a = 43.3 cm, and b = 23.2 cm. Step-by-step solution rounded to one decimal place.

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