Which of the following equations is true by the Law of Sines? \( \frac{\sin B}{23}=\frac{\sin C}{40} \) \( \frac{\sin D}{35}=\frac{\sin B}{40} \) \( \frac{\sin C}{35}=\frac{\sin B}{40} \) \( \frac{\sin B}{40}=\frac{\sin C}{23} \)

To determine which equation is true by the Law of Sines, we must remember that the Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant. This means that for any triangle \( ABC \): \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We can rewrite each option to check for consistency with this rule. The first equation \( \frac{\sin B}{23}=\frac{\sin C}{40} \) implies \( \frac{23}{\sin B} = \frac{40}{\sin C} \), which can be true if there’s a proper triangle. The second option \( \frac{\sin D}{35}=\frac{\sin B}{40} \) can also fit the Law of Sines, depending on angles and the corresponding side lengths. The third equation \( \frac{\sin C}{35}=\frac{\sin B}{40} \) rearranges to a similar format. Finally, \( \frac{\sin B}{40}=\frac{\sin C}{23} \) suggests a different relationship for the angles based on their corresponding sides. After examining all options, it turns out that all equations can potentially fit the Law of Sines depending on specific triangle configurations, provided we know the relationships between the angles and sides. Therefore, without additional triangle information, it is tricky to declare one true above others! Feel free to dive deeper into triangle identities; they’re a universe of their own! Don’t forget, mathematical proofs can be as captivating as a mystery novel! 🌟📚