Determine the remaining sides and angles of the triangle $A B C$.
\[
\mathrm{c}=7 \mathrm{mi}, \mathrm{B}=38.17^{\circ}, \mathrm{C}=34.28^{\circ}
\]
Answer
\(\boxed{\text{Final Answer: The remaining sides and angles of the triangle ABC are } A = 107.55^{\circ}, a = 11.85 \text{ mi}, \text{ and } b = 7.68 \text{ mi}.}\)
Step 1 ：Given that side c = 7 mi, angle B = 38.17 degrees, and angle C = 34.28 degrees in triangle ABC.
Step 2 ：First, we can find angle A by subtracting the given angles B and C from 180 degrees. So, \(A = 180 - B - C = 180 - 38.17 - 34.28 = 107.55\) degrees.
Step 3 ：Next, we can use the Law of Sines to find the lengths of sides a and b. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
Step 4 ：Using the Law of Sines, we can find side a as follows: \(a = c \cdot \frac{\sin(A)}{\sin(C)} = 7 \cdot \frac{\sin(107.55)}{\sin(34.28)} = 11.85\) mi.
Step 5 ：Similarly, we can find side b as follows: \(b = c \cdot \frac{\sin(B)}{\sin(C)} = 7 \cdot \frac{\sin(38.17)}{\sin(34.28)} = 7.68\) mi.
Step 6 ：\(\boxed{\text{Final Answer: The remaining sides and angles of the triangle ABC are } A = 107.55^{\circ}, a = 11.85 \text{ mi}, \text{ and } b = 7.68 \text{ mi}.}\)
