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To solve this triangle using the Law of Sines, we will use the given values:
- Side \( b = 3 \)
- Angle \( A = 40^\circ \)
- Angle \( C = 110^\circ \)

Since the angles in a triangle sum to \( 180^\circ \), we can first find angle \( B \) by calculating:
\[
B = 180^\circ - A - C
\]

### Step 1: Calculate Angle \( B \)
\[
B = 180^\circ - 40^\circ - 110^\circ = 30^\circ
\]

Now, we have:
- Angle \( A = 40^\circ \)
- Angle \( B = 30^\circ \)
- Angle \( C = 110^\circ \)

### Step 2: Apply the Law of Sines
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Since we know \( b = 3 \) and all three angles, we can use the formula to find \( a \) and \( c \).

### Step 3: Calculate Side \( a \)
Using \(\frac{a}{\sin A} = \frac{b}{\sin B}\):
\[
a = b \cdot \frac{\sin A}{\sin B}
\]
\[
a = 3 \cdot \frac{\sin 40^\circ}{\sin 30^\circ}
\]

Calculating this:
\[
a = 3 \cdot \frac{0.6428}{0.5} \approx 3 \cdot 1.2856 \approx 3.86
\]

### Step 4: Calculate Side \( c \)
Using \(\frac{c}{\sin C} = \frac{b}{\sin B}\):
\[
c = b \cdot \frac{\sin C}{\sin B}
\]
\[
c = 3 \cdot \frac{\sin 110^\circ}{\sin 30^\circ}
\]

Calculating this:
\[
c = 3 \cdot \frac{0.9397}{0.5} \approx 3 \cdot 1.8794 \approx 5.64
\]

### Final Answer
- \( a \approx 3.86 \)
- \( c \approx 5.64 \)
- \( B = 30^\circ \)

Let me know if you would like more details, or if you have any questions!

---

**Related Questions**
1. How would the calculations change if angle \( C \) was different?
2. How can we confirm our answers using the Law of Cosines instead?
3. What if side \( a \) was given instead of \( b \)?
4. How would we solve for an angle if only sides were provided?
5. Can the Law of Sines be used in non-acute triangles?

**Tip:** Always check if your angle measures add up to \( 180^\circ \) when solving triangles.

Solve the triangle using the Law of Sines. (Assume b = 3, ∠A = 40°, and ∠C = 110°. Round the lengths to two decimal places.)

Learn how to solve a triangle using the Law of Sines, given side b = 3 and angles ∠A = 40° and ∠C = 110°. This solution includes calculating the third angle and applying trigonometric formulas to find missing side lengths.

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