In triangle \( A B C, m \angle A=45^{\circ}, m \angle B=40^{\circ} \), and \( a=7 \). Which equation should you solve to find \( b \) ? A. \( b^{2}=7^{2}-2(7) b \cos 40^{\circ} \) B. \( \frac{\sin 45^{\circ}}{7}=\frac{\sin 40^{\circ}}{b} \) C. \( \frac{\cos 45^{\circ}}{7}=\frac{\cos 40^{\circ}}{b} \) D. \( \frac{\sin 45^{\circ}}{b}=\frac{\sin 40^{\circ}}{7} \)

To find \( b \), the most appropriate equation to solve is option B: \( \frac{\sin 45^{\circ}}{7}=\frac{\sin 40^{\circ}}{b} \). This is based on the Law of Sines, which states that the ratios of the lengths of the sides of a triangle to the sines of their opposite angles are equal. In this triangle, you know the angle measures and one side length, making this equation suitable for finding the missing side \( b \). Another fun fact: The angles in triangle \( ABC \) add up to \( 180^{\circ} \). Since you already have \( m \angle A = 45^{\circ} \) and \( m \angle B = 40^{\circ} \), you can calculate \( m \angle C \) as \( 180^{\circ} - (45^{\circ} + 40^{\circ}) = 95^{\circ} \). Now, you have all angles, which amplifies your problem-solving toolkit if you encounter any additional side lengths or angles!