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 $abc$ with $\angle a=45^{\circ}$, $\angle b=77^{\circ}$, side $ab=8$, side $bc=a$

Explanation:

Step1: Isolate $a$

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

Step2: Calculate sine values

$\sin(45^\circ)=\frac{\sqrt{2}}{2}\approx0.7071$, $\sin(77^\circ)\approx0.9744$

Step3: Compute numerator

$8\times0.7071=5.6568$

Step4: Divide to find $a$

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

Answer:

$a\approx5.81$