Certainly! Let's solve for the unknown side [tex]\( a \)[/tex] using the Law of Sines. Here is the detailed step-by-step solution:
1. Use the Law of Sines:
[tex]\[
\frac{\sin (A)}{a} = \frac{\sin (B)}{b}
\][/tex]
2. Substitute the given values:
[tex]\[
\frac{\sin (45^\circ)}{a} = \frac{\sin (77^\circ)}{8}
\][/tex]
3. Cross multiply to solve for [tex]\( a \)[/tex]:
[tex]\[
8 \sin (45^\circ) = a \sin (77^\circ)
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{8 \sin (45^\circ)}{\sin (77^\circ)}
\][/tex]
Given the numerical values we have:
[tex]\[
\sin (45^\circ) \approx 0.7071
\][/tex]
[tex]\[
\sin (77^\circ) \approx 0.9744
\][/tex]
Plugging these values into the equation:
[tex]\[
a = \frac{8 \times 0.7071}{0.9744}
\][/tex]
[tex]\[
a \approx \frac{5.6568}{0.9744} \approx 5.8057
\][/tex]
Finally, rounding this to the nearest hundredth:
[tex]\[
a \approx 5.81
\][/tex]
Therefore, the value of [tex]\( a \)[/tex], rounded to the nearest hundredth, is:
[tex]\[
a \approx \boxed{5.81}
\][/tex]
