Question

solve the triangle.

b = □°
(type an integer or a decimal. do not round until the final answer. then round to the nearest degree as needed.)

a = □
(type an integer or a decimal. do not round until the final answer. then round to the nearest tenth as needed.)

c = □
(type an integer or a decimal. 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, \( B=180^{\circ}-(28^{\circ} + 45^{\circ})=107^{\circ} \)

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

The Law of Sines is \( \frac{a}{\sin A}=\frac{b}{\sin B} \).
We know \( A = 28^{\circ}, B=107^{\circ}, b = 18 \).
So, \( a=\frac{b\sin A}{\sin B}=\frac{18\times\sin28^{\circ}}{\sin107^{\circ}} \)
\( \sin28^{\circ}\approx0.4695,\sin107^{\circ}\approx0.9563 \)
\( a=\frac{18\times0.4695}{0.9563}\approx8.8 \)

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

Using the Law of Sines \( \frac{c}{\sin C}=\frac{b}{\sin B} \)
Since \( C = 45^{\circ}, b = 18, B=107^{\circ} \)
\( c=\frac{b\sin C}{\sin B}=\frac{18\times\sin45^{\circ}}{\sin107^{\circ}} \)
\( \sin45^{\circ}\approx0.7071,\sin107^{\circ}\approx0.9563 \)
\( c=\frac{18\times0.7071}{0.9563}\approx13.3 \)

Answer:

\( B = 107^{\circ} \)
\( a\approx8.8 \)
\( c\approx13.3 \)