Question

solve the triangle.

b = \\(\square^\circ\\)
(do not round until the final answer. then round to the nearest degree as needed.)

b \\(\approx\\) \\(\square\\)
(do not round until the final answer. then round to the nearest tenth as needed.)

c \\(\approx\\) \\(\square\\)
(do not round until the final answer. then round to the nearest tenth as needed.)

triangle image: triangle abc with angle a = 46°, angle c = 72°, side a (opposite angle a) = 6, side b (opposite angle b), side c (opposite angle c)

Explanation:

Step1: Find angle B

The sum of angles in a triangle is \(180^\circ\). So, \(B = 180^\circ - 46^\circ - 72^\circ\)
\(B = 62^\circ\)

Step2: Use the Law of Sines to find side b

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(a = 6\), \(A = 46^\circ\), \(B = 62^\circ\). So, \(\frac{6}{\sin 46^\circ}=\frac{b}{\sin 62^\circ}\)
\(b=\frac{6\times\sin 62^\circ}{\sin 46^\circ}\)
\(\sin 62^\circ\approx0.8829\), \(\sin 46^\circ\approx0.7193\)
\(b=\frac{6\times0.8829}{0.7193}\approx\frac{5.2974}{0.7193}\approx7.4\)

Step3: Use the Law of Sines to find side c

Using \(\frac{a}{\sin A}=\frac{c}{\sin C}\), \(C = 72^\circ\), \(\sin 72^\circ\approx0.9511\)
\(c=\frac{6\times\sin 72^\circ}{\sin 46^\circ}\)
\(c=\frac{6\times0.9511}{0.7193}\approx\frac{5.7066}{0.7193}\approx7.9\)

Answer:

\(B\approx62^\circ\), \(b\approx7.4\), \(c\approx7.9\)