https://file.aiquickdraw.com/mathbot/file/1725914249211eo35sz49.jpg

We are asked to solve for \( b \) in this triangle using the Law of Sines. The Law of Sines is given as:

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

From the triangle in the image:
- Side \( a = 8 \) (opposite the angle \( A = 82^\circ \))
- Angle \( B = 55^\circ \)
- We need to find \( b \), the side opposite angle \( B = 55^\circ \).

### Step-by-step Solution:
Using the Law of Sines:

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

Substitute the given values:

\[
\frac{8}{\sin 82^\circ} = \frac{b}{\sin 55^\circ}
\]

Now, we solve for \( b \):

\[
b = \frac{8 \cdot \sin 55^\circ}{\sin 82^\circ}
\]

Let me calculate the value of \( b \).The length of side \( b \), rounded to the nearest tenth, is \( 6.6 \).

Would you like further details or have any other questions? Here are some related questions you might find helpful:

1. How can I apply the Law of Cosines to solve triangles?
2. What happens if I am given only two angles and one side in a triangle?
3. How can I solve a triangle if all sides are known but no angles?
4. Can the Law of Sines be applied in non-right triangles only?
5. How do I calculate missing angles when I know all the sides?

**Tip:** Always check if your angles add up to 180° when solving for missing angles in a triangle to verify correctness.

Solve for side b using the Law of Sines in a triangle with sides and angles given as 8 units, 82°, and 55°.

Learn how to solve for side b in a triangle using the Law of Sines. The triangle has sides of 8 units with angles 82° and 55°. Follow a step-by-step guide to apply the Law of Sines and find the unknown side b, rounded to the nearest tenth.

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