Question

law of sines
(image: triangle with ( angle a = 45^circ ), ( angle b = 77^circ ), side length ( 8 ) (opposite ( angle b )), side ( a ) (opposite ( angle a )))
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 5.80 ) (marked with ( \boldsymbol{\times} ) for incorrect attempt)

Explanation:

Step1: Recall the law of sines formula

The law of sines is \(\frac{\sin(A)}{a}=\frac{\sin(B)}{b}\), where \(A\), \(B\) are angles and \(a\), \(b\) are the lengths of the sides opposite those angles respectively.

Step2: Substitute the known values

We know \(A = 45^{\circ}\), \(B=77^{\circ}\), and \(b = 8\). Substituting into the law of sines formula, we get \(\frac{\sin(45^{\circ})}{a}=\frac{\sin(77^{\circ})}{8}\).

Step3: Cross - multiply

Cross - multiplying gives us \(8\times\sin(45^{\circ})=a\times\sin(77^{\circ})\).

Step4: Solve for \(a\)

To solve for \(a\), we can re - arrange the equation as \(a=\frac{8\times\sin(45^{\circ})}{\sin(77^{\circ})}\).
We know that \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\approx0.7071\) and \(\sin(77^{\circ})\approx0.9744\).
Substitute these values into the formula for \(a\):
\(a=\frac{8\times0.7071}{0.9744}=\frac{5.6568}{0.9744}\approx5.805\approx5.81\) (rounded to the nearest hundredth)

Answer:

\(a\approx\boxed{5.81}\)