To solve for [tex]\( b \)[/tex] using the Law of Sines, let's break down the problem step-by-step.
The Law of Sines states:
[tex]\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides of the triangle opposite to the angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively.
Given the values:
- [tex]\( a = 5 \)[/tex]
- [tex]\( A = 30^\circ \)[/tex]
- [tex]\( B = 45^\circ \)[/tex]
We are to find the length [tex]\( b \)[/tex].
### Step 1: Convert the Angles to Radians
First, convert the angles from degrees to radians since trigonometric functions in calculus typically use radians.
[tex]\[
A = 30^\circ \times \frac{\pi}{180} = \frac{\pi}{6} \text{ radians}
\][/tex]
[tex]\[
B = 45^\circ \times \frac{\pi}{180} = \frac{\pi}{4} \text{ radians}
\][/tex]
### Step 2: Apply the Law of Sines
Using the Law of Sines:
[tex]\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = a \times \frac{\sin B}{\sin A}
\][/tex]
### Step 3: Calculate the Sines
Calculate the sine values:
[tex]\[
\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
\][/tex]
[tex]\[
\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
### Step 4: Plug in the Values and Solve for [tex]\( b \)[/tex]
Now, substitute the values into the equation:
[tex]\[
b = 5 \times \frac{\frac{\sqrt{2}}{2}}{\frac{1}{2}} = 5 \times \frac{\sqrt{2}}{2} \times 2 = 5 \times \sqrt{2}
\][/tex]
Simplifying:
[tex]\[
b = 5 \sqrt{2}
\][/tex]
### Step 5: Evaluate the Numerical Value
Using the approximate value of [tex]\(\sqrt{2} \approx 1.414\)[/tex]:
[tex]\[
b = 5 \times 1.414 = 7.071
\][/tex]
### Conclusion
Hence, the calculated value of [tex]\( b \)[/tex] is approximately [tex]\( 7.071 \)[/tex].
