Question

determine the remaining sides and angles of the triangle abc.
( c = 7 mathrm { mi }, b = 35.37 ^ { circ }, c = 31.28 ^ { circ } )
find the measure of angle a.
( a = 113.35 ^ { circ } )
(type an integer or a decimal)
find the length of side a.
( a = 12 mathrm { mi } )
(round to the nearest mile as needed)
find the length of side b.
( b = square mathrm { mi } )
(round to the nearest mile as needed)

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(c = 7\) mi, \(C=31.28^{\circ}\), \(B = 35.37^{\circ}\), and \(A=113.35^{\circ}\) (already found).

Step2: Solve for \(b\)

From \(\frac{b}{\sin B}=\frac{c}{\sin C}\), we can express \(b\) as \(b=\frac{c\sin B}{\sin C}\).
Substitute the values: \(b=\frac{7\times\sin(35.37^{\circ})}{\sin(31.28^{\circ})}\).
Calculate \(\sin(35.37^{\circ})\approx0.579\) and \(\sin(31.28^{\circ})\approx0.519\).
Then \(b=\frac{7\times0.579}{0.519}=\frac{4.053}{0.519}\approx7.81\).

Answer:

\(b = 8\) mi (rounded to the nearest mile)