Question

determine the remaining sides and angles of the triangle abc.
a = 200 m, a = 34°42, c = 29°46
what is the measure of angle b?
b = 115°32
what is the length of side b?
b = 317 m
(do not round until the final answer. then round to the nearest meter as needed.)
what is the length of side c?
c = \square m
(do not round until the final answer. then round to the nearest meter as needed.)

Explanation:

Step1: Convert angles to decimal degrees

• Convert \(A = 34^{\circ}42'\) to decimal: \(A=34+\frac{42}{60}=34.7^{\circ}\)
• Convert \(C = 29^{\circ}46'\) to decimal: \(C = 29+\frac{46}{60}\approx29.77^{\circ}\)
• \(B = 115^{\circ}32'=115+\frac{32}{60}\approx115.53^{\circ}\)

Step2: Use the Law of Sines

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
We know \(a = 200\), \(A = 34.7^{\circ}\), \(C\approx29.77^{\circ}\)
From \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we can solve for \(c\)
\(c=\frac{a\sin C}{\sin A}\)
Substitute \(a = 200\), \(\sin A=\sin(34.7^{\circ})\approx0.569\), \(\sin C=\sin(29.77^{\circ})\approx0.497\)
\(c=\frac{200\times0.497}{0.569}\)
\(c=\frac{99.4}{0.569}\approx175\)

Answer:

\(c = 175\space m\)