Question

solve the triangle.
b = \square^{\circ}
(do not round until the final answer. then round to the nearest degree as needed.)
b \approx \square
(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.)
triangle with angle a = 45^{\circ}, angle c = 78^{\circ}, side a = 8 (opposite angle a), side b (opposite angle b), side c (opposite angle c)

Explanation:

Step1: Find angle B

The sum of angles in a triangle is \(180^\circ\). So, \(B = 180^\circ - A - C\). Given \(A = 45^\circ\) and \(C = 78^\circ\), then \(B = 180 - 45 - 78 = 57^\circ\).

Step2: Find side b using the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(a = 8\), \(A = 45^\circ\), \(B = 57^\circ\). So, \(\frac{8}{\sin 45^\circ}=\frac{b}{\sin 57^\circ}\). Solving for \(b\), we get \(b=\frac{8\sin 57^\circ}{\sin 45^\circ}\). Calculate \(\sin 57^\circ\approx0.8387\), \(\sin 45^\circ\approx0.7071\). Then \(b=\frac{8\times0.8387}{0.7071}\approx\frac{6.7096}{0.7071}\approx9.5\) (rounded to nearest tenth).

Step3: Find side c using the Law of Sines

Using \(\frac{a}{\sin A}=\frac{c}{\sin C}\), so \(c=\frac{8\sin 78^\circ}{\sin 45^\circ}\). Calculate \(\sin 78^\circ\approx0.9781\). Then \(c=\frac{8\times0.9781}{0.7071}\approx\frac{7.8248}{0.7071}\approx11.1\) (rounded to nearest tenth).

Answer:

\(B = 57^\circ\), \(b \approx 9.5\), \(c \approx 11.1\)