use law of sines t solve the triangle A=5degree 40', B=8degree 15', b=4.8

angle C = 166° 5'

b/sin5°40" = 4.8/sin8°15'
b = 3.303

c/sin166°5' = 4.8/sin8°15'
c = 8.045

To solve the triangle using the Law of Sines, we can use the following formula:

sin(A) / a = sin(B) / b = sin(C) / c

Given:
A = 5° 40' (converted to decimal form = 5.67°)
B = 8° 15' (converted to decimal form = 8.25°)
b = 4.8

Let's find angle C first:

C = 180° - A - B
C = 180° - 5.67° - 8.25°
C ≈ 166.08°

Now, let's use the Law of Sines to find side a:

sin(A) / a = sin(C) / c

Let's plug in the values we know:

sin(5.67°) / a = sin(166.08°) / 4.8

We can rearrange the equation to solve for a:

a = (sin(5.67°) * 4.8) / sin(166.08°)

Using a calculator, we get:

a ≈ 0.410

Therefore, the length of side a is approximately 0.410 units.

To use the Law of Sines to solve this triangle, we need to find the remaining angle and side length. Here's how you can do it:

1. Start by converting the given angles A and B from degrees and minutes to decimal form:
- Angle A = 5° 40' = 5 + 40/60 = 5.67 degrees
- Angle B = 8° 15' = 8 + 15/60 = 8.25 degrees

2. Write down the Law of Sines formula:
a/sin(A) = b/sin(B) = c/sin(C)

3. Now, substitute the given values into the formula:
4.8/sin(B) = b/sin(B) = c/sin(C)

4. We already have b = 4.8, so let's focus on finding the remaining angle and side:
- To find angle C, subtract the sum of angles A and B from 180 degrees: C = 180 - A - B
C = 180 - 5.67 - 8.25 = 166.08 degrees

- Next, find side a using the Law of Sines:
a/sin(A) = b/sin(B)
a = b * (sin(A) / sin(B))
a = 4.8 * (sin(5.67) / sin(8.25))
a ≈ 2.318

5. Now we have all three angles and sides of the triangle:
A ≈ 5.67 degrees
B ≈ 8.25 degrees
C ≈ 166.08 degrees
a ≈ 2.318
b = 4.8
c: We have all the sides, so we don't need to calculate it.

Therefore, the values for the triangle are approximately:
A ≈ 5.67 degrees, B ≈ 8.25 degrees, C ≈ 166.08 degrees,
a ≈ 2.318, b = 4.8, and c is determined with given sides.