Question

solve the following triangle. a = 10°, b = 50°, a = 8 c ≈ □° (simplify your answer.) b ≈ □ (type an integer or decimal rounded to two decimal places as needed.) c ≈ □ (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Find angle \( C \)

The sum of angles in a triangle is \( 180^{\circ} \). So \( C=180^{\circ}-A - B \).
Substitute \( A = 10^{\circ} \) and \( B = 50^{\circ} \):
\( C=180^{\circ}-10^{\circ}-50^{\circ}=120^{\circ} \)

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

The Law of Sines is \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \).
We want to find \( b \), so \( b=\frac{a\sin B}{\sin A} \).
Substitute \( a = 8 \), \( A = 10^{\circ} \), \( B = 50^{\circ} \):
\( b=\frac{8\times\sin(50^{\circ})}{\sin(10^{\circ})}\)
Since \( \sin(50^{\circ})\approx0.7660 \), \( \sin(10^{\circ})\approx0.1736 \)
\( b=\frac{8\times0.7660}{0.1736}\approx35.25 \)

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

Using \( c=\frac{a\sin C}{\sin A} \)
Substitute \( a = 8 \), \( A = 10^{\circ} \), \( C = 120^{\circ} \)
Since \( \sin(120^{\circ})=\frac{\sqrt{3}}{2}\approx0.8660 \), \( \sin(10^{\circ})\approx0.1736 \)
\( c=\frac{8\times0.8660}{0.1736}\approx40.00 \)

Answer:

\( C = 120^{\circ}\), \( b\approx35.25\), \( c\approx40.00 \)