Question

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

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{c}{\sin C}\).

Step2: Substitute the known values

We know that \(c = 7\) mi, \(C=31.28^{\circ}\), and \(A = 113.35^{\circ}\). So, \(a=\frac{c\times\sin A}{\sin C}\).
Substitute the values: \(a=\frac{7\times\sin(113.35^{\circ})}{\sin(31.28^{\circ})}\).
First, calculate \(\sin(113.35^{\circ})\approx0.92\) and \(\sin(31.28^{\circ})\approx0.52\).
Then \(a=\frac{7\times0.92}{0.52}=\frac{6.44}{0.52}\approx12.38\approx12\) (rounded to the nearest mile).

Answer:

\(a = 12\) mi