Question

solve the triangle.
a = 123°, c = 28°, c = 120
b = □°
(do not round until the final answer. then round to the nea
a ≈ □
(do not round until the final answer. then round to th
b ≈ □
(do not round until the final answer. then round

Explanation:

Step1: Find angle B

The sum of angles in a triangle is \(180^{\circ}\). So, \(B = 180^{\circ}-A - C\).
Substituting \(A = 123^{\circ}\) and \(C=28^{\circ}\), we get \(B=180^{\circ}-123^{\circ}-28^{\circ}=29^{\circ}\).

Step2: Use the Law of Sines to find side a

The Law of Sines states \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
We know \(c = 120\), \(A = 123^{\circ}\), \(C = 28^{\circ}\). So, \(a=\frac{c\sin A}{\sin C}\).
\(\sin123^{\circ}=\sin(180^{\circ}- 57^{\circ})=\sin57^{\circ}\approx0.8387\), \(\sin28^{\circ}\approx0.4695\).
\(a=\frac{120\times0.8387}{0.4695}\approx215\).

Step3: Use the Law of Sines to find side b

By the Law of Sines \(\frac{b}{\sin B}=\frac{c}{\sin C}\).
Since \(B = 29^{\circ}\), \(\sin29^{\circ}\approx0.4848\), \(c = 120\), \(\sin C=0.4695\).
\(b=\frac{120\times0.4848}{0.4695}\approx124\).

Answer:

\(B = 29^{\circ}\), \(a\approx215\), \(b\approx124\)