Question

use the law of sines to find the values of ( b ) and ( c ).
( b approx )
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Find angle \( B \)

In a triangle, \( A + B + C=180^{\circ} \). Given \( A = 41^{\circ} \), \( C = 14^{\circ} \), then \( B=180^{\circ}-(41^{\circ}+14^{\circ}) = 125^{\circ} \).

Step2: Apply the Law of Sines for \( b \)

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \( a = 6 \), \( A = 41^{\circ} \), \( B = 125^{\circ} \). From \(\frac{a}{\sin A}=\frac{b}{\sin B}\), we get \( b=\frac{a\sin B}{\sin A} \).
Substitute \( a = 6 \), \( \sin A=\sin41^{\circ}\approx0.656 \), \( \sin B=\sin125^{\circ}\approx0.819 \) into the formula:
\( b=\frac{6\times0.819}{0.656}=\frac{4.914}{0.656}\approx7.5 \)

Answer:

\( 7.5 \)