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 = blank

Explanation:

Step1: Recall the law of sines formula

$\frac{\sin A}{a}=\frac{\sin B}{b}$

Step2: Substitute the given values

We know $A = 45^{\circ}$, $B=77^{\circ}$ and $b = 8$. So, $\frac{\sin45^{\circ}}{a}=\frac{\sin77^{\circ}}{8}$

Step3: Cross - multiply

$8\times\sin45^{\circ}=a\times\sin77^{\circ}$

Step4: Solve for $a$

$a=\frac{8\times\sin45^{\circ}}{\sin77^{\circ}}$
Since $\sin45^{\circ}=\frac{\sqrt{2}}{2}\approx0.707$ and $\sin77^{\circ}\approx0.974$, then $a=\frac{8\times0.707}{0.974}=\frac{5.656}{0.974}\approx5.81$

Answer:

$5.81$