Certainly! Let's use the Law of Sines to solve this problem where we are given:
- Side [tex]\( a = 4 \)[/tex]
- Angle [tex]\( A = 30^\circ \)[/tex]
- Side [tex]\( b = 5 \)[/tex]
We need to find [tex]\( B \)[/tex], an angle opposite side [tex]\( b \)[/tex].
### Step-by-Step Solution:
1. Convert Degrees to Radians:
First, convert angle [tex]\( A \)[/tex] from degrees to radians. The conversion factor is [tex]\( \pi \)[/tex] radians for [tex]\( 180^\circ \)[/tex]:
[tex]\[
A = 30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6} \text{ radians}
\][/tex]
2. Apply the Law of Sines:
According to the Law of Sines:
[tex]\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\][/tex]
Substitute the known values:
[tex]\[
\frac{4}{\sin(30^\circ)} = \frac{5}{\sin B}
\][/tex]
3. Calculate [tex]\(\sin(30^\circ)\)[/tex]:
We know from trigonometry that:
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
Therefore:
[tex]\[
\frac{4}{\frac{1}{2}} = \frac{5}{\sin B}
\][/tex]
4. Solve for [tex]\(\sin B\)[/tex]:
This simplifies to:
[tex]\[
4 \times 2 = \frac{5}{\sin B}
\][/tex]
[tex]\[
8 = \frac{5}{\sin B}
\][/tex]
Rearrange to isolate [tex]\(\sin B\)[/tex]:
[tex]\[
\sin B = \frac{5}{8} = 0.625
\][/tex]
5. Find Angle [tex]\( B \)[/tex]:
The value of [tex]\(\sin B\)[/tex] gives us:
[tex]\[
B = \arcsin(0.625)
\][/tex]
Calculating [tex]\(\arcsin(0.625)\)[/tex] yields:
[tex]\[
B \approx 0.675 \text{ radians}
\][/tex]
6. Convert Radians to Degrees:
Finally, convert angle [tex]\( B \)[/tex] from radians to degrees. The conversion factor is [tex]\( 180^\circ / \pi \)[/tex] radians:
[tex]\[
B \approx 0.675 \text{ radians} \times \frac{180^\circ}{\pi} \approx 38.682^\circ
\][/tex]
So, the angle [tex]\( B \)[/tex] is approximately:
- [tex]\(\sin B \approx 0.625\)[/tex]
- [tex]\(B \approx 0.675\)[/tex] radians
- [tex]\(B \approx 38.682^\circ\)[/tex]
These detailed calculations lead us to the final results for [tex]\(\sin B\)[/tex] and angle [tex]\( B \)[/tex] in both radians and degrees.
