Question

using the law of sines for the aas case
complete the work to determine the value of ( a ).

1. use the law of sines: ( \frac{sin(a)}{a} = \frac{sin(b)}{b} ).
2. substitute: ( \frac{sin(45^circ)}{a} = \frac{sin(77^circ)}{8} ).
3. cross multiply: ( 8 sin(45^circ) = a sin(77^circ) ).
4. solve for ( a ) and round to the nearest hundredth: ( a approx square )

Explanation:

Step1: Recall the formula from step 3

We have the equation \(8\sin(45^{\circ}) = a\sin(77^{\circ})\). To solve for \(a\), we need to isolate \(a\) by dividing both sides of the equation by \(\sin(77^{\circ})\). So the formula for \(a\) is \(a=\frac{8\sin(45^{\circ})}{\sin(77^{\circ})}\).

Step2: Calculate the values of the sines

First, we know that \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\approx0.7071\) and \(\sin(77^{\circ})\approx0.9744\) (using a calculator to find the sine values).

Step3: Substitute the values into the formula for \(a\)

Substitute \(\sin(45^{\circ})\approx0.7071\) and \(\sin(77^{\circ})\approx0.9744\) into the formula \(a = \frac{8\times0.7071}{0.9744}\).
First, calculate the numerator: \(8\times0.7071 = 5.6568\).
Then, divide the numerator by the denominator: \(a=\frac{5.6568}{0.9744}\approx5.80\).

Answer:

\(a\approx\boxed{5.80}\)