To solve for $b$ in the triangle using the Law of Sines, we can follow these steps:

Step 1: Identify the given information.

• Angle $A = 42^\circ$
• Angle $B = 85^\circ$
• Side $a = 31$ (opposite angle $A$)
• We need to find $b$ (opposite angle $B$).

Step 2: Find the third angle of the triangle.

Since the sum of angles in a triangle is always $180^\circ$:

$C = 180^\circ - A - B = 180^\circ - 42^\circ - 85^\circ = 53^\circ$

Step 3: Apply the Law of Sines.

The Law of Sines states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Substitute the known values:

$\frac{31}{\sin 42^\circ} = \frac{b}{\sin 85^\circ}$

Step 4: Solve for $b$.

$b = \frac{31 \cdot \sin 85^\circ}{\sin 42^\circ}$

Now, calculate the values using a calculator:

$b = \frac{31 \cdot \sin 85^\circ}{\sin 42^\circ} \approx \frac{31 \cdot 0.9962}{0.6691} \approx \frac{30.8822}{0.6691} \approx 46.1$

Final Answer:

$b \approx 46.1$

This is the length of side $b$ rounded to the nearest tenth.

Would you like more details or have any other questions?

Here are five related questions to explore:

1. How would you solve for side $c$ using the Law of Sines?
2. What would the triangle look like if angle $A$ was 50° instead of 42°?
3. How does the Law of Cosines differ from the Law of Sines, and when should it be used?
4. What is the relationship between the sides and angles in a right triangle?
5. How can you verify the result obtained from the Law of Sines using another method?

Tip: Always check that the sum of the angles in your triangle equals 180° to ensure accuracy before applying trigonometric laws.