Final answer:
To find the value of 'a' given the angles and one side length using the Law of Sines, we calculate a ≈ 5.80 by substituting the values into the formula and solving.
Step-by-step explanation:
To determine the value of a using the Law of Sines, we are given two angles A and B and the length of one side b. We use the equation provided:
\(\frac{\sin(A)}{a} = \frac{\sin(B)}{b}\)
Substitute the given information into the equation:
\(\frac{\sin(45^\circ)}{a} = \frac{\sin(77^\circ)}{8}\)
Next, cross multiply to solve for a:
\(8 \times \sin(45^\circ) = a \times \sin(77^\circ)\)
Finally, solve for a:
\(a = \frac{8 \times \sin(45^\circ)}{\sin(77^\circ)}\)
Use a calculator to find the approximate value of a, rounding to the nearest hundredth:
a \approx \frac{8 \times 0.7071}{0.9744}
a \approx \frac{5.6576}{0.9744}
a \approx 5.80