Use the Law of Sines to solve the triangle. Round your answers to two decimal places.
B = 31°,  C = 97°,  a = 3
5
8

Given the triangle with the following parameters:

- \( B = 31^\circ \)
- \( C = 97^\circ \)
- \( a = 3.58 \)

We can solve for the remaining angle \( A \) and the sides \( b \) and \( c \) using the Law of Sines.

### Step 1: Find the missing angle \( A \)

Since the sum of the angles in any triangle is \( 180^\circ \):

\[
A = 180^\circ - B - C = 180^\circ - 31^\circ - 97^\circ = 52^\circ
\]

### Step 2: Use the Law of Sines

The Law of Sines states:

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

We know \( a = 3.58 \), \( A = 52^\circ \), \( B = 31^\circ \), and \( C = 97^\circ \). First, we’ll solve for \( b \) and \( c \).

### Step 3: Solve for \( b \)

Using the ratio involving \( a \) and \( b \):

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

Substitute the known values:

\[
\frac{3.58}{\sin 52^\circ} = \frac{b}{\sin 31^\circ}
\]

Now, calculate the sine values:

\[
\frac{3.58}{0.7880} = \frac{b}{0.5150}
\]

Solve for \( b \):

\[
b = \frac{3.58 \times 0.5150}{0.7880} \approx 2.34
\]

### Step 4: Solve for \( c \)

Using the ratio involving \( a \) and \( c \):

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

Substitute the known values:

\[
\frac{3.58}{\sin 52^\circ} = \frac{c}{\sin 97^\circ}
\]

Since \( \sin 97^\circ \approx 0.9986 \), solve for \( c \):

\[
\frac{3.58}{0.7880} = \frac{c}{0.9986}
\]

Solve for \( c \):

\[
c = \frac{3.58 \times 0.9986}{0.7880} \approx 4.53
\]

### Final Answer

- \( A = 52^\circ \)
- \( b \approx 2.34 \)
- \( c \approx 4.53 \)

Would you like more details or further explanations?

### Related Questions:

1. What happens when the Law of Cosines is used instead of the Law of Sines?
2. Can the Law of Sines be used to solve any triangle?
3. How does the Law of Sines differ for obtuse triangles?
4. What are the possible ambiguities when solving triangles with the Law of Sines?
5. How can you verify your solution for a triangle using another method?

### Tip:

Always check if the sum of angles is \( 180^\circ \) before applying any trigonometric laws!

Use the Law of Sines to solve the triangle. Round your answers to two decimal places. B = 31°, C = 97°, a = 3.58

Learn how to solve a triangle using the Law of Sines. This problem involves solving for unknown sides and angles of a triangle with angles B = 31°, C = 97°, and side a = 3.58. Includes step-by-step calculations for angles and sides, using trigonometric formulas. Suitable for high school students in Grades 9-11.

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