Question

part 2 of 2
use the law of sines to find the values of b and c

( b approx 7.1 )
(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)

Explanation:

Step1: Find angle \( B \)

The sum of angles in a triangle is \( 180^{\circ} \). So \( B=180^{\circ}-(43^{\circ}+83^{\circ}) = 54^{\circ} \).

Step2: Use the Law of Sines to find \( b \)

The Law of Sines is \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \). Given \( a = 6 \), \( A=43^{\circ} \), \( B = 54^{\circ} \), \( C=83^{\circ} \). For \( b \), \( \frac{a}{\sin A}=\frac{b}{\sin B}\), so \( b=\frac{a\sin B}{\sin A}=\frac{6\sin54^{\circ}}{\sin43^{\circ}}\approx7.1 \) (already given).

Step3: Use the Law of Sines to find \( c \)

Using \( \frac{a}{\sin A}=\frac{c}{\sin C} \), then \( c=\frac{a\sin C}{\sin A}=\frac{6\sin83^{\circ}}{\sin43^{\circ}} \).
Since \( \sin83^{\circ}\approx0.9925 \), \( \sin43^{\circ}\approx0.6820 \), \( c=\frac{6\times0.9925}{0.6820}=\frac{5.955}{0.6820}\approx8.7 \)

Answer:

\( c\approx8.7 \)