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

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The Law of Sines states that in any triangle, the ratio of the sine of an angle to the length of its opposite side is constant. Mathematically, this is expressed as:

$$
\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
$$

where $$ A $$, $$ B $$, and $$ C $$ are the angles, and $$ a $$, $$ b $$, and $$ c $$ are the lengths of the sides opposite those angles, respectively.

Let's examine each of the provided equations to determine which one aligns with the Law of Sines:

1. **$$ \frac{\sin B}{23} = \frac{\sin C}{40} $$**  
   This equation suggests that angle $$ B $$ is opposite a side of length 23 and angle $$ C $$ is opposite a side of length 40. According to the Law of Sines, this ratio should indeed hold true if the triangle's configuration matches these assignments.

2. **$$ \frac{\sin D}{35} = \frac{\sin B}{40} $$**  
   The presence of angle $$ D $$ introduces ambiguity since a triangle only has three angles (typically labeled $$ A $$, $$ B $$, and $$ C $$). Without additional context, this equation doesn’t clearly correspond to the Law of Sines in a standard triangle setting.

3. **$$ \frac{\sin C}{35} = \frac{\sin B}{40} $$**  
   This suggests that angle $$ C $$ is opposite a side of length 35 and angle $$ B $$ is opposite a side of length 40. While this could be correct in a specific triangle, it conflicts with the first option if both are assumed to be part of the same triangle.

4. **$$ \frac{\sin B}{40} = \frac{\sin C}{23} $$**  
   This is the reciprocal of the first equation. It also holds true but implies a different assignment of sides to angles, which would contradict the first option if both are part of the same triangle.

Given the information and assuming a standard triangle configuration, the first equation aligns correctly with the Law of Sines without introducing conflicting assignments or ambiguity.

**Answer:** 
$$\frac{\sin B}{23} = \frac{\sin C}{40}$$
The correct equation by the Law of Sines is $$ \frac{\sin B}{23} = \frac{\sin C}{40} $$.

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

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