Question

determine the remaining sides and angles of the triangle abc. c = 9 mi, b = 39.31°, c = 33.83° find the measure of angle a. a = □° (type an integer or a decimal.) find the length of side a. a = □ mi (round to the nearest mile as needed.) find the length of side b. b = □ mi (round to the nearest mile as needed.)

Explanation:

Step1: Find angle A

The sum of angles in a triangle is \(180^\circ\). So, \(A = 180^\circ - B - C\).
\(A = 180 - 39.31 - 33.83\)
\(A = 106.86^\circ\)

Step2: Find side a using the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{c}{\sin C}\). So, \(a=\frac{c\sin A}{\sin C}\).
First, calculate \(\sin A\) and \(\sin C\):
\(\sin A=\sin(106.86^\circ)\approx0.9563\)
\(\sin C=\sin(33.83^\circ)\approx0.5578\)
Then, \(a = \frac{9\times0.9563}{0.5578}\approx\frac{8.6067}{0.5578}\approx15.43\approx15\) (rounded to nearest mile)

Step3: Find side b using the Law of Sines

Using \(\frac{b}{\sin B}=\frac{c}{\sin C}\), so \(b=\frac{c\sin B}{\sin C}\).
Calculate \(\sin B=\sin(39.31^\circ)\approx0.6336\)
\(b = \frac{9\times0.6336}{0.5578}\approx\frac{5.7024}{0.5578}\approx10.22\approx10\) (rounded to nearest mile)

Answer:

• Measure of angle \(A\): \(106.86^\circ\)
• Length of side \(a\): \(15\) mi
• Length of side \(b\): \(10\) mi