Question

determine the remaining sides and angles of the triangle abc.
c = 7 mi, b = 37.82°, c = 34.15°
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, \(A = 180^\circ - B - C\).
\(A = 180 - 37.82 - 34.15\)
\(A = 180 - 71.97\)
\(A = 108.03^\circ\)

Step2: Find side a using the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{c}{\sin C}\). So, \(a=\frac{c\sin A}{\sin C}\).
We know \(c = 7\), \(A = 108.03^\circ\), \(C = 34.15^\circ\).
\(\sin A=\sin(108.03^\circ)\approx0.9511\), \(\sin C=\sin(34.15^\circ)\approx0.5623\)
\(a=\frac{7\times0.9511}{0.5623}\)
\(a=\frac{6.6577}{0.5623}\approx11.84\approx12\) (rounded to nearest mile)

Step3: Find side b using the Law of Sines

Using \(\frac{b}{\sin B}=\frac{c}{\sin C}\), so \(b=\frac{c\sin B}{\sin C}\).
\(\sin B=\sin(37.82^\circ)\approx0.6157\)
\(b=\frac{7\times0.6157}{0.5623}\)
\(b=\frac{4.31}{0.5623}\approx7.67\approx8\) (rounded to nearest mile)

Answer:

• \(A = 108.03^\circ\)
• \(a = 12\) mi
• \(b = 8\) mi