Question

consider a triangle abc like the one below. suppose that b=53°, c=82°, and a=6. (the figure is not drawn to scale.) solve the triangle. round your answers to the nearest tenth. if there is more than one solution, use the button labeled \or\. a = □°, b = □, c = □

Explanation:

Step1: Find angle \( A \)

The sum of angles in a triangle is \( 180^{\circ} \). So \( A=180^{\circ}-B - C \).
Substitute \( B = 53^{\circ} \) and \( C = 82^{\circ} \):
\( A=180^{\circ}-53^{\circ}-82^{\circ}=45^{\circ} \)

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

The Law of Sines is \( \frac{a}{\sin A}=\frac{b}{\sin B} \).
We know \( a = 6 \), \( A = 45^{\circ} \), \( B = 53^{\circ} \).
So \( b=\frac{a\sin B}{\sin A} \).
\( \sin 45^{\circ}=\frac{\sqrt{2}}{2}\approx0.707 \), \( \sin 53^{\circ}\approx0.799 \)
\( b=\frac{6\times0.799}{0.707}=\frac{4.794}{0.707}\approx6.8 \)

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

By the Law of Sines \( \frac{a}{\sin A}=\frac{c}{\sin C} \)
\( \sin 82^{\circ}\approx0.990 \)
\( c=\frac{a\sin C}{\sin A}=\frac{6\times0.990}{0.707}=\frac{5.94}{0.707}\approx8.4 \)

Answer:

\( A = 45^{\circ} \), \( b\approx6.8 \), \( c\approx8.4 \)