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:

$a\approx\square$
triangle details: $\angle a=45^{\circ}$, $\angle b=77^{\circ}$, side $b=8$, side $a$ is opposite $\angle a$

Explanation:

Step1: Isolate $a$

$a = \frac{8\sin(45^\circ)}{\sin(77^\circ)}$

Step2: Calculate sine values

$\sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071$, $\sin(77^\circ) \approx 0.9744$

Step3: Compute numerator

$8 \times 0.7071 = 5.6568$

Step4: Divide to find $a$

$a \approx \frac{5.6568}{0.9744}$

Answer:

$a \approx 5.81$