Question

solve each triangle abc that exists.
b = 34°42 a = 38.7 b = 31.2
a = □□° c = □□ c =
(simplify your answer. round to the nearest (rol
degree as needed. round to the nearest as n
minute as needed)
ible solutions for the triangle.
for the solution with the longer side c are as follows.
c₁ = 100°18 c₁ = □
er. round to the nearest (round to the nearest tenth
round to the nearest as needed.)
for the solution with the shorter side c are as follows
c₂ = 10°18 c₂ = □
er. round to the nearest (round to the nearest tenth
round to the nearest as needed.)
c. there are no possible solutions for this triangle

Explanation:

Step1: Convert \( B = 34^{\circ}42'\) to decimal degrees

Since \( 42'=\frac{42}{60}=0.7^{\circ}\), so \( B = 34.7^{\circ}\)

Step2: Use the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\)

We know \(a = 38.7\), \(b = 31.2\), \(B = 34.7^{\circ}\)
\(\sin A=\frac{a\sin B}{b}=\frac{38.7\sin(34.7^{\circ})}{31.2}\)
\(\sin(34.7^{\circ})\approx0.57\)
\(\sin A=\frac{38.7\times0.57}{31.2}=\frac{22.059}{31.2}\approx0.707\)
\(A_1=\sin^{- 1}(0.707)\approx45^{\circ}\) or \(A_2 = 180^{\circ}-45^{\circ}=135^{\circ}\)

Step3: Calculate \(C\) for each case

Case 1: \(A_1 = 45^{\circ}\)

\(C_1=180^{\circ}-A_1 - B\)
\(C_1=180^{\circ}-45^{\circ}-34.7^{\circ}=100.3^{\circ}\)
Convert \(0.3^{\circ}\) to minutes: \(0.3\times60 = 18'\), so \(C_1 = 100^{\circ}18'\)
Use the Law of Sines \(\frac{c_1}{\sin C_1}=\frac{b}{\sin B}\)
\(c_1=\frac{b\sin C_1}{\sin B}=\frac{31.2\sin(100.3^{\circ})}{\sin(34.7^{\circ})}\)
\(\sin(100.3^{\circ})\approx0.983\), \(\sin(34.7^{\circ})\approx0.57\)
\(c_1=\frac{31.2\times0.983}{0.57}=\frac{30.6696}{0.57}\approx53.8\)

Case 2: \(A_2 = 135^{\circ}\)

\(C_2=180^{\circ}-A_2 - B\)
\(C_2=180^{\circ}-135^{\circ}-34.7^{\circ}=10.3^{\circ}\)
Convert \(0.3^{\circ}\) to minutes: \(0.3\times60 = 18'\), so \(C_2 = 10^{\circ}18'\)
Use the Law of Sines \(\frac{c_2}{\sin C_2}=\frac{b}{\sin B}\)
\(\sin(10.3^{\circ})\approx0.179\), \(\sin(34.7^{\circ})\approx0.57\)
\(c_2=\frac{31.2\times0.179}{0.57}=\frac{5.5848}{0.57}\approx9.8\)

Answer:

For the solution with the longer side \(c\): \(c_1\approx53.8\)
For the solution with the shorter side \(c\): \(c_2\approx9.8\)