Question

find the length of side c.
round to the nearest tenth.

Explanation:

Step1: Find angle \( B \)

Sum of angles in a triangle is \( 180^{\circ} \).
\( B=180^{\circ}-72^{\circ}-50^{\circ}=58^{\circ} \)

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

Law of Sines: \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \). Here \( a = 40\), \(A = 72^{\circ}\), \(C = 50^{\circ}\)
\(\frac{c}{\sin50^{\circ}}=\frac{40}{\sin72^{\circ}}\)
\(c=\frac{40\times\sin50^{\circ}}{\sin72^{\circ}}\)
\(\sin50^{\circ}\approx0.766\), \(\sin72^{\circ}\approx0.951\)
\(c=\frac{40\times0.766}{0.951}\approx32.0\)

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

\(\frac{b}{\sin58^{\circ}}=\frac{40}{\sin72^{\circ}}\)
\(b=\frac{40\times\sin58^{\circ}}{\sin72^{\circ}}\)
\(\sin58^{\circ}\approx0.848\)
\(b=\frac{40\times0.848}{0.951}\approx35.6\)

Answer:

\( B = 58^{\circ}\), \(c\approx32.0\), \(b\approx35.6\)