Question

determine the remaining sides and angles of the triangle abc.
a = □° (round to the nearest degree as needed.)
b ≈ □ m
(do not round until the final answer. then round to the nearest hundredth as needed)
c ≈ □ m
(do not round until the final answer. then round to the nearest hundredth as needed)

Explanation:

Step1: Find angle A

The sum of angles in a triangle is \(180^\circ\). So, \(A = 180^\circ - 60^\circ - 59^\circ\)
\(A = 61^\circ\)

Step2: Find side b using the Law of Sines

Law of Sines: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
We know \(a = 14\), \(A = 61^\circ\), \(B = 59^\circ\)
So, \(\frac{14}{\sin 61^\circ}=\frac{b}{\sin 59^\circ}\)
\(b=\frac{14\times\sin 59^\circ}{\sin 61^\circ}\)
Calculate \(\sin 59^\circ\approx0.8572\), \(\sin 61^\circ\approx0.8746\)
\(b=\frac{14\times0.8572}{0.8746}\approx\frac{12.0008}{0.8746}\approx13.72\)

Step3: Find side c using the Law of Sines

Using \(\frac{a}{\sin A}=\frac{c}{\sin C}\), \(C = 60^\circ\)
\(c=\frac{14\times\sin 60^\circ}{\sin 61^\circ}\)
\(\sin 60^\circ=\frac{\sqrt{3}}{2}\approx0.8660\)
\(c=\frac{14\times0.8660}{0.8746}\approx\frac{12.124}{0.8746}\approx13.86\)

Answer:

\(A = 61^\circ\)
\(b\approx13.72\) m
\(c\approx13.86\) m